Proofs of these statements you can find in the usual books, so let me discuss the reason why do we have the absoluteness to begin with.
First of all, the trait of transitive models is that they are substructures of the universe. If $M$ is a transitive model, and $M\models x\in y$ then $x\in y$. This, and the transitivity of $M$ of course, is the key reason why $\Delta_0$ formula are absolute.
Recall that a $\Delta_0$ formula is a formula is equivalent to a formula written using bounded quantifiers, e.g. $\varphi(x)$ being $(\exists y\in x)(\forall z\in y)(z\notin x)$. But bounded quantification pass between the transitive models exactly because they are transitive, if $M\models(\exists y\in x)(\forall z\in y)(z\notin x)$ then this statement holds for the real $\in$ relation, and it holds for all the members of $x,y,z$ and so on.
On the other hand, if $V$ satisfies $\varphi(x)$, and $x\in M$ then by transitivity it is not hard to see that $y$ which exists also lies in $M$, and that every $z\in y$ also lies in $M$, and therefore $M$ satisfies this formulas as well.
Formally we prove this by induction on the number of quantifiers, or length, or complexity of the formula. Whatever you find the easiest, neither is very difficult.
The reason for $\Delta_1$ absoluteness is slightly trickier, but it's a very nice trick indeed. Let $\varphi(x,y)$ be a $\Delta_0$ formula, if for some $x\in M$ we have that $M\models\exists y\varphi(x,y)$ then $M$ knows about a witness for $\varphi(x,y)$. But now we have this witness in the universe, and $\varphi(x,y)$ is a $\Delta_0$ formula so $\varphi(x,y)$ holds in the universe, so $\exists y\varphi(x,y)$ is true in the universe.
Similarly, if $\forall y\varphi(x,y)$ holds in the universe, and $x\in M$ then it holds for all $y\in M$ as well, and $\varphi(x,y)$ is again a $\Delta_0$ formula so it is absolute and $M\models\varphi(x,y)$ for every $y\in M$, and therefore $M\models\forall y\varphi(x,y)$.
So existential quantification goes up and universal quantification goes down. But what is a $\Delta_1$ formula? It is a formula which is equivalent to an existential quantification of a $\Delta_0$ formula, and equivalent to a universal quantification of a[nother] $\Delta_0$ formula. One is absolute upwards and the other downwards making our $\Delta_1$ formula absolute in both directions.