Let $f : X \to C$ be a surjective map between projective varieties ($C$ is a curve). Let $C^* = C - \{\text{critical values of $f$}\}$, $X^* = f^{-1}(C^*)$. Fix $t \in C^*$ and let $X_t = f^{-1}(t)$.
There are inclusions $i : X_t \to X^*$ and $j : X^* \to X$ which induces map in singular cohomology : $i^* : H^m(X^*) \to H^m(X_t), j : H^m(X) \to H^m(X^*)$.
1) I saw that "obviously" $H^m(X_t)^{\pi_1(C_t)} = i^*H^m(X^*)$. Why is this true ?
2) The invariant cycle theorem states that $i^*j^* H^m(X) = H^m(X_t)^{\pi_1(C_t)}$. What is the geometric intepretation of this statement ? What are the interesting corollaries ?
This is just an answer to 1) and some comments.
As I said in the comment, the main point is Leray spectral sequence and its functoriality. So Leray spectral sequence is $$ E_2^{pq}=H^p(C^*,R^qf_*\mathbb{C})\Rightarrow H^{p+q}(X^*) $$ Concretely this means that we can recover the cohomology of the total space if we know the cohomology of the base $C^*$, the sheaves $R^qf_*\mathbb{C}$ and a lot a differentials between the different groups. These differential may be very hard to track, but the functoriality maybe of tremendous help here.
In the present situation, $f:X^*\rightarrow C^*$ is a proper submersion. This implies that $R^qf_*\mathbb{C}$ are local system whose fibers are $H^q(X_t)$. From the general theory of local system, we have $$H^0(C^*,R^qf_*\mathbb{C})=H^q(X_t)^{\pi_1(C^*,t)}$$ What about other values of $p$ ? This is where we will use functoriality : the map $i^*:H^{p+q}(X^*)\rightarrow H^{p+q}(X_t)$ is compatible with a morphism of spectral sequence $$ i^*:H^p(C^*,R^qf_*\mathbb{C})\rightarrow H^p(\{t\},R^qf_*\mathbb{C}|_t)$$ But the last group is zero unless $p=0$ in which case it is $H^q(X_t)$. It follows that the image of $i^*:H^q(X^*)\rightarrow H^q(X_t)$ is exactly the image of $i^*:H^0(C^*, R^qf_*\mathbb{C})=H^q(X_t)^{\pi_1(C^*,t)}\rightarrow H^q(X_t)$.
A word about 2). The Leray spectral sequence is a very powerful to understand the cohomology of a space. But it is often very hard to compute explicitly the differentials, so you can use tricks as above to have more informations. Now from Hodge theory, there is huge restriction on the the differentials : classes in the spectral sequence have weights and the differential preserve them. It turns out that most of the time, they will be zero because they map spaces of different weights. For example we have the very important and powerful theorem that if $f$ is projective smooth between algebraic varieties, then the Leray spectral sequence degenerates at $E_2$ : every single classes survives the spectral sequence and $H^{n}(X^*)$ is actually isomorphic to $\bigoplus_{p+q=n}H^p(C^*,R^qf_*\mathbb{C})$.
Weights behave this way : if $X$ is projective then $H^m(X)$ has weights $\leq m$ whereas if $X$ is smooth $H^m(X)$ has weights $\geq m$. Thus if $X$ is projective and smooth $H^m(X)$ is pure of weight $m$.
But $i^*:H^m(X^*)\rightarrow H^m(X_t)$ preserves weights, so every classes of weight $>m$ is mapped to $0$. We also have $j^*:H^m(X)\rightarrow H^m(X^*)$ and this morphism preserve weights, so every classes of weights $<m$ is mapped to $0$. With a bit more work, both $H^m(X)$ and $H^m(X^*)$ have the same part of weight $m$, and as we said before, this is the part which have a non trivial image in $H^q(X_t)$. Hence the global invariant cycle theorem.