The degree of vertex $i \in V$ is a number of edges adjacent to $i$. First question asks you to proof that the sum of degrees of all vertices from left side of the bipartite graph is equal to the sum of all degrees of all vertices from right side. The intuition is that every edge that starts in the left side has to end somewhere in the right side. Moreover the question asks you to do it by induction on the number of edges in the graph.
First show that if there are no edges in the graph then the equality holds (indeed $0 = 0$, doesn't it?). Then, assuming that for every graph with $n$ edges the equality is true, show that it is also true for every graph with $n+1$ edges (so we have a bipartite graph with $n+1$ edges, then remove any edge, the equality holds, so if you add the edge in question back, and the graph is bipartite, you add $1$ to both sides of the equation, so equality holds again).
A 103-regular graph is a graph where every vertex has degree of 103, i.e. every vertex has 103 adjacent edges. You need to show that if graph is $103$-regular, then $|A| = |B|$, that means the number of vertices on the left side is the same as in the right (the intuition is that if you have $n$ vertices on the left with degree of 103, then you have $103 n$ edges from left to right, and those edges have to end somewhere, and the vertices on the right side has degree of 103 too, so you need $\frac{103 n}{103}$ vertices on the right side).
This of course can be done by induction, but it is easier to show directly.
I didn't write the whole proof by intention, it is almost everything, but you should reformulate it into mathematical notation. Hope that solves your problem.
P.S. Considering the notation in the question I guess you deal with math or computer science. You have said that "Now worse comes graph theory which I barely [...] see any application for". I don't want to be rude, but if I were you I would force myself to learn to love graphs. Graphs are omnipresent in mathematics and in other domains too, you will meet them eating breakfast, driving work, learning, working, talking to people, shopping or flying, even your brain is a graph! Please, for you own good, work hard to understand the basics and even harder to see some applications (those are especially important)! On a happier note, graphs are (in some way) indeed beautiful, and believe me, knowing them you can appreciate how marvelous our world is even more. Good luck!