Let's take these in reverse order since it seems like a more natural progression:
- (2.2) Where and when will you apply it?
Homotopy theory is part of algebraic topology. At a high level we use it to map topological and geometric questions into algebraic ones. By doing so you can leverage algebraic techniques which may help answer your geometric question. For example, when you have two different topological spaces on your hands and you are trying to see if they are topologically different you can try to calculate the homotopy groups and compare. When you can establish a homotopy between two spaces we say they're homotopic. Certain geometric properties are preserved by homotopy equivalence and you can often use this information to answer your original geometric question.
- (2.1) Why do we need to define homotopy between two continuous functions?
Homotopy formally captures the act of 'deformation'. You deform one thing into another continuously and say they're homotopic. Usually people are trying to calculate the fundamental group associated with the topological space (or geometric object). The idea is to pick a point on the thing (called the base point) and then start making loops all over the space that start and end on the base point. Those loops are continuous maps from the interval [0,1] into the space and those are the continuous functions that are getting deformed into each other with a homotopy. Go crazy and draw them all over the space starting from the base point and then see if you can shrink (or retract) them back to the original point or deform one loop into another. If you can deform one loop into another then you say there exists a homotopy between them and can consider them equivalent. You define an operation between these equivalence classes of paths and you get something with a group structure -- the fundamental group. In essence you have distilled out an algebraic structure from the space and now have something you can calculate with and compare with that of other spaces.
(1.1) Doesn't homotopy preserve the area/volume of the objects?
This question doesn't really make sense because the whole idea is to formalize the notion of deformation. You deform one thing into another and it plows over the whole idea of metric. No metric no volume or area... so when you think homotopy toss the whole area/volume thing out. By using homotopy you're trying to extract algebraic information from the space that didn't care about things like area or volume. From the homotopy point of view a big doughnut and a sphere are different (one's got a hole but the other don't) but two doughnuts of different sizes are the same. It's the hole that mattered in the end... not the relative sizes of each doughnut.
HTH and sorry if the I used language that was too loose. :) It sounded like you were a comp-sci guy so I was trying to be a bit less formal.