Good question! This has to do with what it means to "feel heavy." Usually, we say that the weight you feel is your "apparent weight." Lots of people think that apparent weight is just how hard gravity is pulling you down, but that's not true. Think about it this way: if you were in orbit around the Earth, you feel weightless (your apparent weight is zero), but that doesn't mean gravity isn't pulling on you! The Earth still pulls on you, keeping you in orbit.

Your apparent weight is really how hard the ground is pushing you up. When you're just standing on the Earth, the ground is pulling you up just as hard as the Earth's gravity is pulling you down, so the ground's push and the Earth pull cancel, and you don't move. But what about when you're in an elevator? Well, if the elevator starts moving up, then you need to start moving up along with it (when I say "moving up," I really mean "accelerate upwards"). But that means that the floor needs to push you extra hard to make you start moving up - that makes you feel heavier. When the elevator stops speeding up and moves at a constant speed, the floor doesn't need to you to speed up anymore, so you feel your normal weight.

I'm under the impression you mean, why do we feel heavier in a lift/elevator (when going up)? This is due to gravity and the force of moving upwards quickly. Gravity is the force that makes items fall toward the ground. If you hold a ball, and then suddenly let go, it falls toward the ground right? Gravity has been measured, such that if you drop an object (no matter how much it weighs - think bowling ball vs. pencil) from a really, really, really tall building, everything falls at the same rate. On Earth this value is 9.81 m/s^2 - or if you drop an object that is not initially moving, it will fall 9.81 m in the first second, 19.22 m in the second second, etc. Now, if we think about moving away from the ground (or moving toward the atmosphere), we feel both the force of gravity, and the force of moving. The force of moving upwards is just like moving forward in a car. Think of sitting in a car, and having it start quickly - we are pushed backward in our seats, and we feel the force of moving forward. Instead of moving forward in a lift/elevator, we move upwards, so we feel ourselves being pushed downwards, making ourselves feel heavier. If we were on the moon in an elevator, there would be less gravity, and we wouldn't feel nearly as heavy in the first place. We would however feel slightly heavier still when the elevator moves up. You can also think about the opposite feeling, like feeling lighter when an elevator moves downwards quickly (faster than the force of gravity).