>Can anybody explain what it is about monarchies that the Iranian regime is so opposed to?

Perhaps if the monarchy hadn't been, or hadn't been seen as, the puppet of foreign interests, Iranians wouldn't have had such a dislike for it. I'm not sure that the type of government Iran had mattered so much as how it operated. An autocratic government is one thing, but subjecting people to that, political/religious repression, inflation, corruption, and violence made revolution inevitable. If the government had ostensibly been a democracy, the result would have been the same, and the succeeding rulers would have denounced democracy in a similar fashion.

>I find this confusing because it seems to me that the present-day Iranian theocracy is very similar to a monarchy itself.

Superficially, yes, and you can argue that perhaps the theocracy is merely a means to an end for some who seek power, but I don't think it's possible to claim that the systems are the same because you have observed one similarity. In a monarchy a ruler can, within some unofficial limitations, make whatever decisions they wish and are not accountable to a higher authority, nor do their decisions have to come from, or be in line with, precedent. In a theocracy, religion informs the rule of the state, and there are conventions, lines of reasoning, and established precedent for virtually all decisions that do not equate to "Because I say so." from a supreme ruler. Whether it is done sincerely, or as an excuse, is another matter.

The impression I've gotten from the past few decades is that a big chunk of the Iranian people hate being a theocracy (or being close to one,) as well, so there probably isn't much of an actual difference in perception.

The author and the date are both significant: Khomeini went far further than most of the religious leadership in his opposition to the Shah's personal rule, and here he's simultaneously proclaiming the illegitimacy of the then regime and the need for clerical leadership of a new state. It's really the moment when the outlines of the post-1979 order are first laid out.

But it wasn't always always thus: the clergy had held the Safavid dynasty in high regard (reciprocating its promotion of clerical authority), and even after viewing its successors as usurpers, senior religious figures made their peace with the Shah after the 1953 coup before their falling-out in the 1960s gave the already outspoken Khomeini his opportunity to claim spiritual leadership of the opposition movement.

Nor was Khomeini's authoritarian clericalist take characteristic of past anti-regime religious sentiment, senior religious figures often siding from the 1890s with popular protest movements (to some extent foreshadowing 1979) and being associated with the 1906-11 constitutionalist movement and (at least for a time) the parliamentary cause in the early 1950s. The republic's eventual form was in part a historical accident, Khomeini emerging just as an earlier generation of leaders was passing from the scene.