There are a lot of analogies but they all fail in some way. I think PBS Spacetime does the best in general, with good graphics to back up the words.

My layman's explanation is probably all stuff you've heard before. Massive objects "warp" spacetime and things that get stuck in those "wells" eventually fall to the bottom due to drag (from a variety of sources).

You've also probably seen the rubber sheet with a bowling ball in the middle used to represent that warping. To visualize that in 3D, I like to imagine a 3D grid of nodes and edges (like a jungle gym of joints and bars) where the whole thing is flexed inward towards a center point. More warped near the center, less warped further out. That kind of conveys the acceleration from gravity felt by things around that center mass.

That's a pretty massive problem.

Hehe guess what.... Nobody exactly knows how gravity works.

So let's start with Newton. According to his equations, gravity is a force. Why? Well, according to the first law of motion, an object stays in motion in its original direction till a force is applied on it. Newton said, "Gravity is clearly actively changing the direction of an object. Hence, gravity is a force". As Newton establishes that gravity is a force, we say that it "pulls" objects. However, he couldn't explain the mechanism behind this pulling force.

Then came Einstein. According to general relativity, gravity is not a force. How's that possible? Doesn't gravity change the motion of a given object? Nope! Wait, whaaaaa?! Okay, so according to special relativity, all things in the universe are on a "spacetime". What's a spacetime? Well, it's a four dimensional fabric like thingey that all objects are present on. The four dimensions are time and the three spatial dimensions that we experience. Mathematically, there is no difference between time and the spatial dimensions. However, all objects move only in one direction through spacetime at c, ie., lightspeed through spacetime. If an object moves at c through space, it moves at 0 m/s through time and vice versa. This is how you get the time dilation magic. Cool. Now what if we bent this said spacetime at certain points? The object traveling on this spacetime would be traveling in a straight line always. Hence, no force. Hence, no pulling. Hence, no pulling mechanism.

Cool! So we solved gravity, right? Sike! GR doesn't work at the quantum level...... Aaaaand most of the best models that we have for quantum gravity use a particle called the "graviton", which has a field that results in an attractive force. How does this force work? The answer is "go fk urself".

Hence, in conclusion, noone knows whether gravity even "pulls" in the first place, let alone HOW it pulls. Aaaaand we've been trying to answer this question for almost 100 years..... Cool....

I've heard some people say the gravitational fields push instead of pull. I still don't get it either.