I agree: the sentence "second number after $3k$: $3k + 1 + 2$, or $3k + 3$" is extremely misleading. They change the meaning of $k$ mid-sentence.
I think to understand what is going on we should make as clear as possible the difference between the small numbers we add up in each step and the big triangular numbers.
The situation they start from is that for some small number (such as 6) that is itself divisible by 3 we find a big triangular number (in the example: 21) that is also divisible by 3. Now we want to introduce the number $k$ to write numbers that are divisible by 3 as 3k. The author of the picture does not specify which of the two they rewrite in this way and this remains unclear forever so let's do it better ourselves.
Let's say the small number (6 in our example) is denoted 3k (so k = 2 in our example) and the big number (21 in our example) I will write 3K (so the big number K is 7 in our example).
Now the next triangular number is formed by adding the next small number (3k + 1) to the big number 3K, so we get $3K + 3k + 1$. This can be rewritten as $3(K + k) + 1$ so this number is one more than a multiple of 3 and hence not a multiple of 3 itself. It is of the form $3m + 1$ (with m = K + k) but saying that it is 3k + 1 as the picture does is extremely misleading.
Now the next triangular number we get by adding the next small number: $3k + 2$ and so the next triangular number is $3K + (3k + 1) + (3k + 2)$. This can be expanded to $3(K + k + k) + (1 + 2)$, a number of the form $3n + 3$ (just take $n = K + k + k$) and hence divisible by $3$ as the picture argues. However calling this number $3k + 3$ is serious abuse of notation.
The next triangular number we get by adding the next small number $3k + 3$ to the total we already had and hence we end up with $3(K + k + k) + 3 + (3k + 3) = 3(K + k + k + k) + 6 = 3(K + k + k + k + 2)$ so again this number is divisible by 3.
I hope this helps!
The final thing you need to understand is that after three steps we are back in the same situation we started with:
The last small number we added (3k + 3) is divisible by 3 and the large triangular number we arrived at, 3(K + k + k + k + 2), also.
Since these two properties was what defined our starting position we can repeat the same reasoning over and over.