The suggested intuition behind mixed Hodge structures - developed 
in particular to generalize Hodge decomposition of cohomology 
groups from smooth and complete Kähler manifolds to more general 
algebraic varieties - is that one should think the cohomology groups
$H^k(X)$ to be endowed with increasing filtrations
whose successive quotients originate from  
cohomologies of appropriate smooth complete varieties, 
hence admit (pure) Hodge structures, *but* of different weights. 



Here is an motivating and so rather 'informal' example which I took from here: [Example of curves][1]  
where I missing this decisive feature that 
successive quotients of the associated filtration shall come from the 
cohomologies of smooth *complete* varieties.

The example works as follows:

>To motivate the definition, consider the case of a reducible complex 
algebraic curve $X$ consisting of two nonsingular components, $X_1 $ and $X_2$,
 which transversally intersect at the points $Q_1$ and $Q_2$. 
Further, assume that the components are not compact, but can be compactified by adding 
the points $P_1 , ... , P_n$. The first cohomology group of the curve $X$ (with compact support)
is dual to the first homology group, which is easier to visualize. 
There are three types of one-cycles in this group. First, there are elements 
$ \alpha_{i}, (i=1,..., n)$ representing small loops around the punctures 
$P_{i}$. Then there are elements $ \beta_{j} $  that are coming from the 
first homology of the compactification of each of the components. 
The one-cycle in $ X_{k}\subset X$  ( $ k=1,2$ ) corresponding to a cycle in the 
compactification of this component, is not canonical: these elements are determined 
modulo the span of $\alpha_{1} ,... , \alpha_{n}$. 
Finally, modulo the first two types, the group is generated by a combinatorial cycle 
$\gamma $ which goes from $ Q_{1}$ to $ Q_{2}$ along a path in one component 
$X_{1}$  and comes back along a path in the other component $X_{2}$. 
This suggests that $ H_{1}(X)$ admits an increasing filtration

$$ 0\subset W_{0}\subset W_{1}\subset W_{2}=H_{1}(X) $$

>whose successive quotients $W_n/W_{n−1}$ originate from the cohomology of smooth 
complete varieties, hence admit (pure) Hodge structures, albeit of different weights. 

*Question:* The last point I not understand. From cohomology of which concrete *smooth complete varieties* originate the cycles $ \alpha _{i}$ generating $W_0$ as claimed there?  
The same question about the combinatorical cycle $\gamma $ generating $W_2/W_{1}$.




  [1]: https://en.wikipedia.org/wiki/Hodge_structure#Example_of_curves