Argthtjtr

As far as I understand, Theresa May was elected to the leader of the Conservatives (and consequently Prime Minister of the UK) to make a Brexit deal (source):

She [May] said there was a "big job" ahead to unite the party and the country following the referendum, to "negotiate the best possible deal as we leave the EU" and to "make Britain work for everyone".

She added: "I am the only candidate capable of delivering these three things as prime minister[...]"

She worked for two years on a deal that the Parliament eventually rejected, so she failed her main(?) task. Yet, the next day the very same Parliament gave her confidence that she should continue governing (and achieve a Brexit deal). How do MPs explain that on Tuesday they reject her deal, but on Wednesday they trust her to continue to get an acceptable deal (which she failed to do in two years, according to the vote the previous day)?

That is because there is a likelihood that the opposition party will gain power should the government be defeated in the confidence vote.

According to the Fixed-term Parliaments Act 2011, the government has 14 days to try to form a new government or an early election will have to be called. The new government formed will also be subjected to a confidence vote.

The Act specifies that early elections can be held only:

[ ... ]

if a motion of no confidence is passed and no alternative government is confirmed by the Commons within 14 days.

The Conservative Party is currently in power on a "confidence and supply" arrangement with the Democratic Unionist Party.

It would be difficult for the Conservative Party to find another party to form a "confidence and supply" arrangement. After the 2017 general elections, the Liberal Democrats expressed skepticism on forming a government with the Conservative Party, the Scottish National Party is opposed to the Conservative Party while the Sinn Féin has an abstentionist policy. These are the three parties with enough seats to prop up the government.

After the damage inflicted on the Liberal Democrats by their coalition deal with the Conservatives in 2010-15, the centrist party ruled out any reprise. There was also no chance of a Conservative deal with the Scottish National Party (SNP), which won 35 seats but which is resolutely opposed to the Tories on both constitutional and economic questions. It appears that no one has even contemplated a grand coalition between Labour and the Conservatives, an arrangement that works in Germany but which is alien to the UK other than in wartime.

^{Source: The Conservation: Can a minority Conservative government survive? Let’s look at the maths}

It's also worth noting that it is rare for a party's own MPs to vote against their government in a confidence motion.

Most governments are defeated after the "confidence and supply" party (in the current case, the DUP) votes against it. However, the DUP is opposed to a Corbyn government so they continue to prop up the incumbent Conservative Party government.

Not surprisingly, MPs voted entirely along party lines on the confidence motion:

If no new government could be formed, an early election must be held in which the Labour Party is currently in good shape to win.

As such, either way, the opposition Labour Party will likely gain power should the incumbent Conservative Party be defeated in the confidence vote.

Because, I think, no one wants to be the Prime Minister place during Brexit. Just imagine the number of problems coming - with borders, economy, and the bank sector especially. For now, responsibility for all Brexit-coming problems will lie on the May's government. On the next elections, May's government will be associated with all Brexit stuff, not the new-elected.

MPs don't like May's Brexit deal. But they also don't want to take responsibility for the consequences, by substituting May's government themselves.

As far as I understand, Theresa May was elected to the leader of the Conservatives (and consequently Prime Minister of the UK) to make a Brexit deal

No, she was elected leader of the Conservatives (and consequently Prime Minister of the UK) to lead the Conservatives (and consequently the country). Making a Brexit deal is surely the most important part of those jobs at the moment so it is still somewhat surprising that she is still in them. However, your premise suggests a direct connection between the Brexit deal and her election, which simply does not exist.