What is the evidence for the existence of gravity
Etymology
If you look in a dictionary, particularly in one that shows etymology (origin of words) you will see that gravity is an old word meaning weight.
It also has the meaning of seriousness (e.g. the notion that serious ideas or subjects weigh heavily on the mind) - but we can ignore that.
Apples
It is everyone's everyday experience that if you let go of an apple, it will tend to move towards the ground.
Science is mostly about measuring this sort of phenomenon and working out how long it takes to move the distance from hand to ground, how fast it travels in each part of the journey and whether the time taken depends on colour, smell, density, day of week, holder's hat-size etc.
So far you shouldn't have any difficulty in believing in
- the notion that things have weight
- the notion that bigger things tend to be heavier
- the notion that heavier things are things which are harder for you to keep from moving from your hand to the ground
- the notion that some materials have more weight than their size would suggest (small iron cannonballs are heavier than similar sized tennis balls)
You should have access to plenty of evidence that all the above are true.
You can just name this tendency of heavy things to move as, their gravity.
Theory of gravity
Many people of a scientific bent studied the apple phenomenon. They hoped to come up with some mathematical way of describing the way apples move. Then you could use the mathematical description to also predict how coconuts and other objects would move. Military types thought this might help them with moving their cannonballs onto the heads of people they didn't like very much.
To scientists a "theory" is the name they give for any sort of mathematical description that actually works to describe something in ways that can be usefully used to predict what happens next.
Newton came up with a theory about how heavy stuff moves (i.e. about stuff with gravity)
Newton's law of universal gravitation states that every mass attracts every other mass in the universe, and the gravitational force between two bodies is proportional to the product of their masses, and inversely proportional to the square of the distance between them.
The notion that the Earth attracts apples shouldn't be too hard to find evidence for. Apples dropped in Australia move in a different direction than apples dropped in Canada, but both move towards the large heavy object we call Earth.
For the part about the inverse square law. You can perform all sorts of tests but one way is to apply the theory to the motion of planets about stars or moons about planets and see how well the theory's predictions of what happens next matches what you actually see happening next.
But note that Newton's theory is a description for two objects only.