I couldn't understand the following concept:"two expression are compatible if one of them can be obtained by adding some expression to the right end of the other."(Shoenfield Mathematical Logic, page 15). For example, $ab$ and $cd$ are compatible then $a$ and $c$ are compatible.
Can someone explain this more clearly please?
The two expressions "abcd" and "abcdefg" are compatible because you can talke the former and add "efg" to the right side, and it turns into the latter. THe expressions "abcd" and "abd" are not compatible, since no matter what you put on the right side of any of them, they won't become the same expressions.
In your case, the letters $a, b, c$ and $d$ are placeholders for (potentially very long) expressions, so we need some care in handling them. But if $ab$ and $cd$ are compatible, then there is some expression $e$ such that either $abe$ is the same expression $cd$, or such that $ab$ is the same as $cde$. I'll assume the latter is the case, the former goes completely analoguously.
If $ab$ is the same expression as $cde$, write them up next to eachother, and delete one letter from the right end of each of them. Repeat this deletion until either all that's left of $ab$ is $a$, or all that's left of $cde$ is $c$, whichever comes first.
Now, if both of them ocurred simultaneously, then $a$ and $c$ are the same expressions, and thus compatible. If $a$ was left alone before $cde$ had been completely reduced to $c$, then whatever remains to the right of $c$ is a string of symbols to the right of the $c$ part is exactly what you need to put to the right of $c$ to make it identical to $a$, and therefore $a$ and $c$ are compatible. Vice versa if $c$ is the first expression to be left alone.