Rav Shraga Yair, the son of Rav Nosson Dovid of Shidlovtza zy"a (Sefer Aron Eidus), explains this pasuk in accordance with the Gemara (Chulin 2B) that says that one shouldn't make a vow by saying, "I accept upon myself to bring a Korban Olah" because he might be negligent and fail to fulfill his vow. Instead, he should say about a specific animal that "harei zeh Olah. (This is an Olah.)" This presents no concern as he has already separated and sanctified the animal, so his vow is already fulfilled.
This idea applies to all areas of life. Whenever one has an arousal to do any mitzvah and he commits to do it, he should not push it off for a later time. He shouldn't say that "I accept upon myself to do it" because he may never end up doing it. Rather, one should fulfill "harei zeh" by getting it done right away.
Accordingly, the pasuk can be learned as saying that Moshe told the leaders of the nation to teach the people that "this is the thing (zeh hadavar) that Hashem has commanded." They should tell the people that Hashem commanded them to do "zeh", meaning to fulfill mitzvos right away, rather than merely committing to do them later, and thereby potentially allowing their enthusiasm to wane.
This is also hinted to in the words "lo yachel devaro" (he shall not violate his words, Devarim 30:3). The word "yachel" is related to the words "mayachalim" (those who hope, as in Tehillim 33:18). Thus, the pasuk can be understood as saying that when one commits to do a good deed, he should not wait and hope to do it later. Rather, "whatever comes out of his mouth, he shall do" immediately, while he still retains his original enthusiasm.
