The weak form of the PDE means that you have, in some sense, "fewer constraints on the kinds of functions that you can use" to construct a solution.
To put it into perspective, recall that in FEM, you look for a solution of the form:
where $u_j$ are the unknown scalar coefficients (which will be solved for) and $\phi_j$ are some known functions. The main question is: what kind of functions $\phi_j$ are we allowed to choose?
Well... after going through a very long, tedious, and mathematically pedantic proof, we arrive at a very nice conclusion which is very convenient for engineers who want to implement FEM:
All $\phi_j(x)$ must have a value of zero on the boundary.
If the weak form of the PDE has a weak derivative of maximum order $k$, then it is sufficient that the functions $\phi_j(x)$ have continuity of order $k-1$.
Condition #1 is very easy to understand: $\phi_j(x)=0$ on all points along the boundary of the domain of your problem.
Condition #2 is not entirely obvious (also not 100% mathematically or pedantically correct either; just "close enough for engineers' sakes"). But condition #2 is the most useful part because it reveals the continuity constraints on $\phi_j$. For example, take the poisson equation in weak form (assuming homogeneous dirichlet boundary conditions)
$$\int_\Omega k\nabla u \cdot \nabla v dx=\int_\Omega fv .$$
This equation has a weak derivative of maximum order k=1 because the gradient here is, effectively, a first order weak derivative (if the weak form had a laplacian operator $\nabla^2$, then k=2; etc...). Thus, you can choose a function space such that $\phi_j(x)$ has continuity of order zero. In other words, you can use functions that are continuous, but not necessarily smooth, across the entire domain. This gives you a lot of flexibility in being able to construct a set of functions statisfying condtion 1.
Why does this matter at all? Because, in FEM, we also seek a third desireable (albeit not absolutely necessary) condition:
- We want the stiffness matrix to be as sparse as possible.
This third condition matters because it implies that we should use functions $\phi_j$ that are not only zero on the boundary, but zero "almost" everywhere except in small subregions of the domain. Each subregion corresponding to a given $\phi_j$ should only overlap with a small number of other subregions corresponding to other functions.
We can use this condition while exploiting with condition #2 to produce a desirable choice of functions $\phi_j$. Lagrange polynomials are one such set of functions which satisfies all three conditions. Of course, Lagrange polynomials are not the only choice, but are probably the simplest to understand from a beginners' perspective. These functions are zero everywhere except in a small region where they are piecewise linear, quadratic, cubic, etc... Because $\phi_j(x)$ are only non-zero in small regions for each given $j$, the integrals forming the stiffness matrix are also mostly zero and thus the stiffness matrix will also be very sparse.
I just want to caution you that I've watered down a lot of mathematics here and that Condition #2 is only "approximately correct enough for an engineer's perspective" and is not 100% mathematically correct. I leave it to the abundance of mathematically inclined users on this site to point out how gross an overstatement condition #2 is in reality.