Ok, so you have a sequence of some abelian groups or R-modules (more generally objects in some abelian category). These are called the sheets or pages of your spectral sequence. In most cases these are bigraded and visualised by these 2d grids. In these cases, you have three indices, the page, the horizontal, and the vertical index.
Furthermore, every page has an endomorphism whose square is 0. These are called the differentials. Because its square is zero, the image of the differential lies in its kernel. So, you can take the quotient Kernel / Image (the homology).
Now, in a spectral sequence you have that the nth page is the homology of the (n-1)th page.
So, in theory if you know the nth page and the differential of the nth page you can calculate the (n+1)th page.
Also, like in the picture you posted, the differentials get "steeper" with every page. Now, the most spectral sequences (with bigraded objects) have on every page a large area where all terms are zero. So, if you fix the vertical and horizontal index, then at some page all following differentials that end or begin at the term with the fixed indices, are zero. And the term doesnt change anymore when you go forward with the pages. So, every term gets stationary at some point, and you have a infiniteth page (or limit page).
So, what do you want to do with spectral sequences? Mostly, you have a special kind of spectral sequence that occurs in a certain context and a theorem that tells you what the second page and the limit page are. And, the second page is some object that you already know and the limit page is some object that you want to calculate.
So, you begin with the second page and mostly through formal calculations you calculate the third etc.
Why formal calculations? Well, in practice what the differential explicitly does is inaccessible. But, similarly to long exact sequences you can calculate some terms without knowing the maps beforehand.
Any more specific questions?
