This is regarding Theorem 3.9, Chapter 4, A course in functional Analysis by J.B Conway.
I wanted to know that as $U_1$ is a open neighbourhood of zero, why would there exist a continuous seminorm $p$ on $X$ such that $\{x\in X: p(x)<1\}\subset U_1$? And why $B+U\cap A+U$ an empty set?
The continuous seminorm in question is simply the Minkowski functional: if $U$ is a convex subset of $U_1$ containing $0$ then it’s interior is simply the set of points with $p(x)<1$.
For your other question: if $a+u_1=b+u_2$ with $u_{1,2}\in U$ and $a\in A,b\in B$ then $b+u_1-u_2=a$ with $p(u_1-u_2)\leq p(u_1)+p(u_2)<1/2+1/2=1$ which means $u_1-u_2\in U_1$ and thus $B+U_1\cap A\supseteq \{a\}$ contradiction