Transfroming 2-factor to another 2 factor graph. I'm reading Petersen's DIE THEORIE DER REGULAREN GRAPHS.
There is interesting statement at chapter 7, page 6.

Given 2-factor F.
You can transfrom F to any other 2-factor applying following operation many times.
Erase 2 edges ab, cd of F. Create 2 edges ac, bd or ad, bc.

The proof looks short. However, unfortunetly, the paper is written in German. I could not understand the proof with dictionary.
Does anyone explain the proof to me?

I don't think the statement you are asking about is even claimed on p.6 (p.198). Later, on p.7 (p.199) Petersen claims that the edge swap operation lets us turn any graph into any other graph with the same degree sequence. (Maybe he is only claiming this for regular graphs, since I think for Petersen only regular graphs exist...?) This implies the statement you want.
More is true in that regard: even if we do not allow multigraphs to appear in intermediate steps (as Petersen does) then we can turn any graph into any other graph with the same degree sequence, regular or not. This is a theorem proved independently by Havel and by Hakimi, and I go into some detail about how to prove it here. (This gets me out of reading more German, which is not my favorite language for doing math in either.)

Meanwhile, on the page you're looking at, Petersen claims that

If we start with a graph that has a $2$-factorization, and apply this operation, we get another graph that has a $2$-factorization.

There are two cases to check here, which Petersen does. First, if $ab$ and $cd$ are in the same $2$-factor, then deleting them and replacing them by $ac$ and $bd$ leaves degrees in that $2$-factor unchanged, so it is still a $2$-factor, and the $2$-factorization is still preserved.
Second, if $ab$ and $cd$ are in different $2$-factors, the union of those $2$-factors is $4$-regular: a $4$-factor, I suppose, though that's not a very modern thing to say. After applying the edge swap, the $4$-factor is preserved, and by Petersen's preceding theorem, that $4$-factor can be factored into two $2$-factors, restoring the $2$-factorization of the whole graph.
