First statement is true. Yes, spread happens with masks. But it is much lower than without masks.
Second statement is potentially false: If you reduce the rate of spread (with vaccines, masks, etc), then while there is still spread, each person eventually is infecting less than 1 other person on average.
So if you have 1,000 infected people... and they only infect another .8 people each.... then the "next" generation is they infect only 800 people. Then that generation infects only 640... then 512.. then 410, then 328, then 262, then 210, then 169, 134, 107, 86, 69, 55, 44, 35, 28, 23, 18, 14, 12, 9, 7, 6, 5, 4, 3, 2.4, 1.9, 1.5, 1.2, 1, .8, .6, .5, less than a half of a person (so eventually becomes zero).
This is how herd immunity works. You eventually get to a point where each person, on average, is infecting less than 1 other person. Masks and other mitigation, done correctly, can have the same effect. Eventually, the number is driven down to 0.
This example was with a R0 (reproduction rate) of 0.8. The goal of mass vaccination is to get to a R0 rate of about 0.4. At 0.4:
It would be 1,000 --> 400 --> 160 -->64 -->26 --> 10 -->4 --> 2 --> 1 -->0.
So you'd go from 1000 cases to 0 cases.. assuming a 2 week cycle, it would take about 2-3 months to get down to 0 cases.