The sum of two words with carries suppressed is just a convoluted way of saying XOR. You don't need to implement any kind of complicated summation operation. Just perform a bitwise-exclusive OR. I have no idea why Bernstein is being so abstruse. No cryptographer who would possibly be reading his paper would need a refresher on something as ubiquitous in that field as a disjunction.
If you're performing modular addition, just like with paper-and-pen arithmetic, you need to use carries. When adding binary you're doing the same but each digit is taken modulo 2 instead of modulo 10.
1 1 ⟵ carries
0 0 0 1
0 0 1 1 +
⸻⸻
0 1 0 0
We see that decimal 1 (binary 0001) added to decimal 3 (binary 0011) equals decimal 4 (binary 0100). That is, 1 + 3 = 4. So, what happens if we drop the carries, i.e. perform carryless addition in binary?
0 0 0 1
0 0 1 1 ⊕
⸻⸻
0 0 1 0
This is the same as decimal 1 XORed with decimal 3, and the result is decimal 2 (0010)! So 1 ⊕ 3 = 2. Salsa20 does the same thing, but with 32-bit words instead of 4-bit words. Because the addition is modulo 232, the leftmost carry, if there is one, is always suppressed (there isn't space for a 33rd bit).