The short answer is that such threats suggest wider party disapproval over government policy. In principle Mrs May could ignore them, (indeed, more so that most Prime Ministers, having just won a leadership vote meaning she can't be challenged for a year) but only by accepting a danger of defeat in parliamentary votes, which might (if continued unchecked) eventually force a general election, due to the minority status of the Conservative Party in the current parliament.
There are three possible ways "resign" could be interpreted here:
- A Minister could threaten to resign from the Cabinet (theoretically privately, but in practice, this kind of thing always seems to leak). I'm assuming it's actually this one you are talking about.
- An MP of the government's party could threaten to resign the whip (i.e. quit the party and act independently), as has happened recently to both the Conservatives and the Labour party.
- an MP could threaten to "resign" from parliament completely (the process isn't actually called resignation for historical reasons, but there is a mechanism to stop being an MP).
In the first case, the UK parliament has a doctrine known as "collective responsibility", in which cabinet ministers are expected to publicly support (and vote for) the Cabinet's position on all issues, even if they don't personally agree with it. As such, a Minister who threatens to resign is really saying "I feel strongly enough about this issue that I'm going to give up running my department in order to be able to say this is wrong". If the matter is such that other backbench MPs of the party in government feel the same, then it's likely that the Government will lose that vote. Further, the minister will need a replacement. Since all large UK political parties are formed of coalitions of groups with different (but related) ideologies, the PM is left with the difficulty of either finding an MP from the same wing as the one who resigned who is still willing to serve, or risking party unity by appointing someone from another wing of the party.
In the second case, then the split runs further than a single issue, and the MP may start voting against the government on multiple issues. This is particularly relevant at the moment since Mrs. May's administration is a minority government. There are more MPs from the other parties summed together than there are Conservative MPs, as such to guarantee wins in parliament, votes MPs from other parties must be gained. If the defeat were in a special vote called a vote of confidence, then there might have to be a general election.
The third option is actually in some ways the least damaging. Although the government would be down one vote temporarily, a by-election to replace the old MP would be called, and would usually return a new Conservative MP, since most UK seats strongly favour a particular party.