Ttykuu

The controversial backstop is seen by the hard-line Brexiteers as a trap by the EU to keep Northern Ireland permanently in a customs union without an option to get out if the UK wanted. In a no deal scenario, both the Irish PM and the EU presidents have said there would still be no hard border between Ireland and Northern Ireland.

So why do they insist on having it in the withdrawal agreement?

For the EU, keeping a border open even if it should be closed from a strictly legal and commercial point of view will impose costs. These might include the import of non-certified products into the internal market, illegal immigration, and more. So they would rather keep the border open under conditions which permit it, i.e. with a "somewhat hard" border between Northern Ireland and the rest of the UK.

I understand why the latter option would be unacceptable from an Ulster Unionist viewpoint, but from an EU viewpoint it solves lots of problems -- the border between the Republic of Ireland and Northern Ireland is open, but the land on the other side of the open border follows common market regulations.

O.m. is correct, but in a more prosaic representation, relative to no-deal, the backstop pushes the single-market border from the interior of Ireland to the Irish Sea. This is not entirely correct, because the whole UK would be in a more limited customs union with the EU as well. So there's basically a "double pushback" that spreads out the pain of dealing with the customs checks (some get pushed to the Irish Sea, some get pushed even further to the whole UK). With no-deal, all those checks have to be performed somewhere in Ireland (even if not right at the border), where they incur a higher cost for EU, mostly in terms of risks for stirring violence or at least more nationalism in Ireland etc.

Of course the EU needs to hang some carrot for the UK to buy this (as the UK would incur a cost) or at least a stick (worse economic outcome for no-deal). So some of the rest of the Agreement does the carrot part, but it's besides the point of your question, which is limited to EU's motivation for the backstop.

As for "hellbent" that has more to do with EU showing solidarity with Ireland, negotiating tactics, a German sensitivity to borders and Troubles etc.

Germany’s foreign minister, Heiko Maas, was speaking as Ireland’s deputy prime minister, Simon Coveney, said MPs who were planning to vote against Theresa May’s deal needed to stop their “wishful thinking” that the EU would reopen Brexit negotiations.

“Some people call us stubborn, but the truth is avoiding a hard border in Ireland is a fundamental concern for the EU, a union that more than anything else serves one purpose – to build and maintain peace in Europe,” said Maas. [...]

“During the Brexit negotiations, all 27 member states agreed on a common position and stood by it. This unity includes full solidarity with Ireland. We insisted, and still do: a hard border dividing the Irish island is unacceptable." [...] said Maas in a speech to Irish ambassadors in Dublin on Tuesday.

Basically the backstop guarantees a softer border in Ireland compared to no-deal.

Not everyone in Europe was happy with this, Poland in particular raised the idea of a "5-year limited backstop" back in January; Czaputowicz, their foreign minister said:

“Obviously, that would be less beneficial for Ireland than an indefinite backstop, but much better than a no deal Brexit which is unavoidably coming our way”.

According to what Czaputowicz underlined during his comments, the EU has become hostage to Ireland’s government position in the negotiations. “The Irish also gave a pretext to treat the British harshly. Arguably, they thought the UK would at some point agree to an indefinite backstop,” but this did not happen. “Now we have a game of chicken with two cars heading towards each other,” added Czaputowicz, who said this will inevitably mean a process that will lead to a hard border.

but they went back in line quickly after they were disavowed by Germany, Ireland (of course) and the by Brussels. Which as we now now has a contingency plan for the border that isn't that hard, but still riskier (for the EU) than the backstop would be.

The dominant idea in the EU, as expressed by the Irish foreign minister:

“I made it very clear that putting a time limit on an insurance mechanism, which is what the backstop is, effectively means that it’s not a backstop at all. I don’t think that reflects EU thinking in relation to the withdrawal agreement.”

I guess that you look at it from the German perspective, in which the backstop equals peace (at least declaratively), "peace for 5 years" sounds silly. Of course "peace forever or war now" is also not a great slogan. But this is probably where the game of chicken comes in play; the more drastic terminology you use to frame the problem, the more likely it is for the no-deal to seem out of the question.

And this is actually not too far from the actual game of chicken, in which a strategy is

Pre-commitment

One tactic in the game is for one party to signal their intentions convincingly before the game begins. For example, if one party were to ostentatiously disable their steering wheel just before the match, the other party would be compelled to swerve. This shows that, in some circumstances, reducing one's own options can be a good strategy.