I believe the terms meet and join come from lattice theory. A lattice, after all, can be defined as a partially ordered set in which any two elements have a meet and a join. In practice, a lattice typically arises as a collection of "closed" sets (with respect to some kind of algebraic closure) ordered by set inclusion; typical examples would be the lattice of all subgroups of a group, or the lattice of all subspaces of a vector space.
Consider the lattice of subspaces of a vector space. The meet of two subspaces is their set-theoretic intersection; e.g., for two $2$-dimensional subspaces of $\mathbb R^3$, their meet is the line where the two planes meet. The join of two subspaces is what we get when the two subspaces join together to make a bigger subspace; in general it's not just the set-theoretic union, but the linear span of the union.
You also wanted to know where the symbols $\vee$ and $\wedge$ come from. I don't know but I'd guess they are derived from the symbols $\cup$ and $\cap$ for union and intersection, the lattice operations in the lattice of all subsets of a set. As for the symbols $\cup$ and $\cap$, my wild guess is that they are stylized versions of the letters u (for union) and n (for intersection). And if that's not the true history, it's good enough for a mnemonic, isn't it?