If you're not really understanding the definitions of a limit, it might help with a metaphor.
When I do proofs, I think of myself pulling pizzas out of an oven (I used to work in a pizza shop). Think of there being an "ideal" level of cooking for your pizza. Obviously, it is not going to be possible to get this "ideal" amount of cooking for every pizza (or possibly even any pizza), but there is a certain "tolerance" you can have for over-cooking or under-cooking before you consider it raw or burnt. As long as you are reasonably close to the right amount of time needed, then your level of cooking will be considered acceptable. Then as you gain more experience, you should be able to get closer and closer to keeping the pizzas in the oven for the ideal amount of time, thereby making your pizzas closer and closer to the ideal level of cooking, which means you would expect that your tolerance would decrease as you'd be getting used to your pizzas being cooked properly.
So if we were to call the amount of time in the oven , then the level of cooking is some function of x . We said there is an ideal level of cooking, we could call that , which means there is a point in time which gives this ideal level of cooking. Remember we said that as long as we have kept the pizzas in the oven for an amount of time reasonably close to , say units of time away from it, then our level of cooking would be considered acceptable, or within some tolerance which we could call . So we need to show that and are related, so that you are guaranteed that as you get experience and keep your pizzas in the oven closer to the right amount of time ( i.e. gets smaller) then so will your tolerance get smaller and closer to the ideal level of cooking.
Do you see now what it means to show ? It means if you have set a tolerance around your ideal limiting value, then as long as you are reasonably close to , then you are guaranteed that your function value is within your tolerance away from the limiting value, and by showing the relationship between and , you are guaranteed that as your gets smaller and you close in on , then your tolerance will get smaller and your will close in on .