Here is a $\Delta$-complex structure of $\mathbb R\text P^2$ found on the internet:
I find it very difficult to see why this is homeomorphic to the sphere $S^2$ with antipodal pairs identified. I want to start from a simplicial structure of $S^2$ and do some identifications, but it doesn't quite work out. How can I know if this square is homeomorphic to $\mathbb R\text P^2$?
I then try the seemingly simpler case $\mathbb R\text P^1$, but I find that I cannot do the following identification on $S^1$ represented by two vertices and two edges, because it is against the definition of $\Delta$-complexes.
How to find a $\Delta$-complex structure on $\mathbb R\text P^1$?
Instead of seeing directly why it's the sphere with antipodal points joined together, you should note that if you do join antipodal points, each equivalence class is going to contain a point either in the upper hemisphere or else on the equator. So let us view $\mathbb RP^2$ as a quotient of this upper half sphere plus equator. No equivalence class can contain two points in the upper hemisphere, and so points collapse if and only if they are antipodal points points on the equator.
By projecting onto the $xy$-plane, this closed hemisphere is homeomorphic to the unit disc, so we are taking the unit disc and associating antipodal points. But the closed unit disc is homeomorphic to $[0,1]^2$, and associating opposite points on the boundary of the unit square yields exactly the $\Delta$-complex that you drew.