Why do only fixed points contribute to the Euler characteristic?
See [BB], in particular Corollary 2:
Suppose $G=\mathbb{G}_m$ acts on an algebraic scheme $X.$ Then $\chi(X) = \chi (X^G).$
I'm not certain what is precisely meant by an algebraic scheme in that paper, but it probably includes your notion of a variety. Bialynicki-Birula's result is for any algebraically closed field.
If you ever yearn for the same result for any reduced and irreducible scheme over an arbitrary field, see Theorem 1.3 of [HU].
You may also be interested in this blog post by Brendan Pawlowski, which gives a proof sketch of the following:
If a torus $T$ acts smoothly on a compact smooth manifold $M$ with finitely many fixed points, then $|M^T| = \chi(M).$
What I wrote above are the most general results that I am aware of. More usually (in my particular experience) one needs to use information specific to the situation at hand to push through the argument in some way. The end of user160609's answer sketches one approach.
Here are some facts that may help you in a particular situation you encounter.
See section 3.34 of [HA] for an introduction to cohomology with compact supports. The main things you need to know about it are:
If $X$ is a compact space then $H_c^{\bullet}(X) = H^{\bullet}(X).$
$H^i_c(\mathbb{R}^n)= 0$ if $i\neq n$ and has rank $1$ if $i=n.$
Cohomology with compact supports is not homotopy invariant.
If $X$ is homeomorphic to $Y$ then $H^{\bullet}_c(X) \cong H^{\bullet}_c(Y).$
Let $U\subseteq X$ be an open subset and $C= X\setminus U.$ There is a long exact sequence: $$ \cdots \to H^i_c(U) \to H^i_c (X) \to H^i_c (C) \to H^{i+1}_c (U) \to \cdots$$
- For complex algebraic varieties, $\chi_c(X) = \chi(X).$
Proof: If $X$ is a compact space then every singular simplex trivially satisfies the extra condition of compact support, so 1) follows. Property 2) can be found in Section 3.34 of [HA]. Since $\mathbb{R}$ is homotopy equivalent to a point, this shows that cohomology with compact supports is not homotopy invariant. Recall that $X$ is homotopy equivalent to $Y$ if there are continuous maps $f:X\to Y, g:Y\to X$ such that $f\circ g$ is homotopic to the identity on $Y$ and $g\circ f$ is homotopic to the identity on $X.$ The root of the problem is that given a continuous map $X\to Y,$ the induced map on their cohomology does not always restrict to their cohomology with compact support (an example of this is the constant map from $\mathbb{R}^n$ to $\{ 0 \}$). This would be true if the preimage of every compact set in $Y$ was guaranteed to be compact, which is the case when the map is a homeomorphism. The rest of the usual proof that singular cohomology is homotopy invariant goes through and property 4) is still true.
Property 5) is proved is the special case of 7.8 (page 186) of [IV] where the chain map is the identity.
Property 6) is proved on pages 141-142 of [FU].
If $Y$ is locally closed in $X$ then $\chi_c(X) = \chi_c(X\setminus Y) + \chi_c(Y).$
Proof : Suppose $U$ is an open subset of $X$ and $C=X\setminus U.$ Using exactness and the Rank-Nullity theorem at every point in the long exact sequence for cohomology with compact supports (property 5 above) gives $$\sum_{i=0}^{\infty} (-1)^i \dim H^i_c(X) = \sum_{i=0}^{\infty} (-1)^i \left(\dim H^i_c (C) +\dim H^i_c (U)\right),$$ which gives $\chi_c(X) = \chi_c(U) + \chi_c(C).$ We repeatedly apply this property. The closure $\overline{Y}$ of $Y$ is closed, so $\chi_c(X) = \chi_c(X\setminus \overline{Y}) + \chi_c(\overline{Y}).$ Since $Y$ is locally closed (open in its closure) we have $\chi_c(\overline{Y}) = \chi_c(\overline{Y}\setminus Y) + \chi_c(Y).$ Since $X\setminus \overline{Y}$ is open in $X\setminus Y$ we have $\chi_c(X\setminus Y) = \chi_c( X\setminus \overline{Y} ) + \chi_c( \overline{Y}\setminus Y ).$ Combining these gives the result.
Here's an example of how these facts help us to compute the Euler Characteristic. Even if you haven't studied toric varieties, you'll be able to see what type of specific facts we need to prove to push the computation through.
If $X=X(\Delta)$ is a toric variety then $\chi_c(X)$ is the number of nondegenerate cones in $\Delta.$
Proof : First we index the cones of $\Delta$ as $\sigma_1, \sigma_2, \ldots, \sigma_{|\Delta|}$ such that their dimensions are nondecreasing. Let $\mathcal{O}_i$ be the orbit associated to the cone $\sigma_i.$ A fact about toric varieties is that we have a decomposition of $X=X(\Delta)$ into finitely many disjoint locally closed subvarieties: $\displaystyle X= \bigsqcup_{i=1}^{|\Delta|} \mathcal{O}_{i}.$ A theorem about the orbits of the torus action further implies that $\overline{\mathcal{O}_{k+1}}$ is disjoint from $\bigcup_{i=1}^{k} \mathcal{O}_{i}$ since $\sigma_{k+1}$ can not be a face of any of the cones that come before in the sequence. This implies that $\overline{\mathcal{O}_{k+1}},$ the closure of $\mathcal{O}_{k+1}$ in $X,$ is also the closure of $\mathcal{O}_{k+1}$ in the subspace $X\setminus \bigcup_{i=1}^k \mathcal{O}_{i}.$ Therefore $\mathcal{O}_{k+1}$ is still locally closed even when considered as a subset of $X\setminus \bigcup_{i=1}^k \mathcal{O}_{\sigma_i}.$ Applying the result that if $Y$ is locally closed in $X$ then $\chi_c(X) = \chi_c(X\setminus Y)+\chi_c(Y)$ repeatedly gives
$$ \chi_c(X) = \chi_c( \mathcal{O}_1) + \chi_c (X\setminus \mathcal{O}_1) = \sum_{i=1}^2 \chi_c( \mathcal{O}_i) + \chi_c (X\setminus \bigcup_{i=1}^2 \mathcal{O}_i ) = \ \cdots \ = \sum_{i=1}^{|\Delta|} \chi_c(\mathcal{O}_i).$$
Another fact about the orbits is that $\mathcal{O}_i \cong (\mathbb{C}^*)^{\operatorname{codim}(\sigma_i)}.$ Using the facts about cohomology with compact supoorts and the Kunneth formula lets you compute $\chi_c( (\mathbb{C}^*)^n)$ which finishes the computation.
[BB] A. Bialynicki-Birula, On fixed point schemes of actions of multiplicative and additive groups. Topology 12 (1973), 99–103.
[FU] W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, New Jersey, 1993, The William H. Roever Lectures in Geometry.
[HA] A. Hatcher, Algebraic topology. URL: http://www.math.cornell.edu/~hatcher/AT/AT.pdf
[HU] W. Hu, On additive invariants of actions of additive and multiplicative groups. arxiv/0912.0563
[IV] B. Iversen, Cohomology of Sheaves, Universitext Series, Springer-Verlag, Berlin, 1986.