I’ve heard that only a slightly stronger gravitational pull would make it impossible for rockets to launch. Is this true? I’ve heard this used as the reason why humanity is meant to be in space.
There's no "bright line" at which space travel would become impossible; a slightly stronger gravitational pull would require bigger and more expensive rockets. Linear increases in gravity require exponential increases in the size and expense of the rocket, so at some point it becomes impractical1. At some point there's a theoretical barrier (no material exists that you can build a rocket of the required size out of, for example) but the practical engineering and resource limits kick in much earlier than that.
For a planet with twice the surface gravity of Earth, for example, you need a rocket about 90 times the mass of the Atlas launchers used for Project Mercury just to get one person into low planetary orbit. That's 4 times the mass of the Saturn V; beyond that point I don't think most civilizations would even try it.
I’ve heard this used as the reason why humanity is meant to be in space.
Humanity isn't meant to do anything except what humanity decides to do.
1 This may seem intuitively strange, but consider that the more fuel you add, the heavier the rocket is, and so adding 50% more total thrust involves adding much more than 50% more fuel (and thus overall rocket size). This is already a significant mass penalty under Earth gravity, so increases in gravity would make this issue more glaring. For more, read about the Tsiolkovsky Rocket Equation.
As this article points out, rockets quickly get impractical. For example, at 10 times earth gravity, the rocket's mass is comparable to the planet's mass, so that's definitely some sort of limit!
But who said we have to use rockets? Suppose we build a monorail completely encircling the planet at some convenient height $h$ above the ground, and accelerate a vehicle until it's actually in orbit at height $h$ (plus a tiny bit). Then we can use this as a launch platform. Once we're in orbit, albeit at a ludicrously low height, we can use that to maneuver into a higher orbit without using vast amounts of fuel. I mean, this neglects air resistance, and the danger to the rest of the population is of XKCD-like proportions, but on the right planet ...
A (very high) upper limit is defined by the thrust to weight ratio of the first stage engine itself. The engine without a tank and a payload would not be able to lift off if the thrust is smaller than its weight measured under the high gravity.
An engine build for such an extreme gravity would need more structal weight than at Earth's gravity. The atmospheric pressure of a planet with extreme gravity would be very high and reduce the engines exhaust velocity and thrust.
If we define the ratio of engine mass, structural mass, fuel mass and payload mass as well as engine thrust for a hypothetical first stage, we may calculate the maximum gravity for this stage to take off. Payload mass of the first stage would be all other stages total mass plus the spaceship.
Gravity will not keep a species out of space, although it can make it incredibly expensive. A resource-limited species might not be able to make it to space, though--I'm thinking of Jovians.
Chemical rockets suffer the tyranny of the rocket equation, if you need more than 30km/sec to attain orbit I don't think you're doing it, period. However, that's not the only way to space.
user6030 brought up nuclear rockets--nuclear thermal doesn't have the thrust but nuclear pulse (aka Project Orion) does, although there is some question if the pusher plate can be kept from melting. (Before it got scrapped by the atmospheric test ban treaty it got as far as confirming the basic idea--you can use a nearby nuclear detonation for propulsion and survive. What is not answered is if you can keep the plate cool enough in the face of repeated detonations.) Using fusion bombs you can get an ISP of nearly 8,000 -- nearly 20x what you can get from chemical rockets and thus letting you lift off from a world with an escape velocity of perhaps 1000 km/sec.
However, there are three other approaches I'm aware of that have no limits whatsoever other than you must be in a world with chemistry (they might not suffice to get you off a neutron star) in order to build them. All are megaengineering on a scale beyond anything the human race has done to date.
First, and easiest, the launch loop. Build two stations, they lob iron bars back and forth. You need some ginormous magnets to turn them around but no super materials. You build an evacuated tunnel for them, then start flinging them faster and faster--above orbital velocity. Your tunnel is basically a maglev track upside-down--instead of the train riding the track the track rides the train of flying bars. Lift enough of the track out of the atmosphere, then put another linear motor on top to launch from.
Second, the space fountain. Same basic idea but you have only one station, it throws the bars straight up and you have a series of platforms that extract energy from the bars heading up and transfer it to the ones going back down. You have to build to synchronous altitude, then just push off and you're in orbit.
Finally, my own design. Adam Chalcraft sort of touched on it but his is nowhere near a complete solution. Build an evacuated tunnel around the world, supported on pillars. Once again, pieces (or perhaps a solid object in this case) moving at above orbital velocity, riding a track on the top of the tunnel. Spin this until the outward force matches the weight of the tunnel and it's pillars--the net downward force should be zero. Now, do it again on top of the first one. Unlike a building where each floor must be able to support all the floors above, in this case each layer is supported by the spinning weight. The bottom ones have no greater load than the top ones. Repeat until you're out of the atmosphere, then you can launch with a linear motor.
(A simple proof this works: Take it to the infinite extreme--an infinite number of pillars and zero space between the rings. While it can't actually be built that way it should be obvious the forces involved go to zero in this case. Thus the only question is how close together do they need to be given the limits of the construction materials.)
In Russell Borogove's answer, they assert "Linear increases in gravity require exponential increases in the size and expense of the rocket, so at some point it becomes impractical." That is the physics answer, but it's slightly different from an economics perspective. A more precise statement would be that if the only variables are payload and gravity, then payload decreases exponentially with gravity. But they are not the only variables. As the cost grows, there is more and more pressure to decrease the cost. There are many places where more cost-effectiveness could be squeezed out. The US continues to launch from Cape Canaveral, despite equitorial launch sites being optimal. Research into nuclear power has been stymied by safety concerns. And so on. This is very much the realm of unknown unknowns, but it's very likely that a civilization on a planet with gravity significantly greater than Earth's could, if properly motivated, engage in space travel.
If gravity is higher, the density of the atmosphere would be higher as well, and probably also to a greater altitude as well as less would have evaporated into space. So hydrogen or helium balloons would rise up more rapidly or lift more weight to possibly higher altitudes. Maybe this would be the main way of accessing space travel on this imaginary planet.