The English word "overseas" is used to refer generally to other countries, or parts of the world. This stems from the fact that historically the majority of English speakers, when going to another country, travelled over the sea. In Italy, and other parts of Continental Europe, the way to other countries was more often over the mountains than across the sea. The word ultramontes, over the mountains, was thus originally etymologically equivalent to overseas, as a way of referring to abroad, or foreign.
In the Eleventh Century there was a series of disputes between the Holy Roman Emperor Henry IV, in Germany, and Pope Gregory VII (Hildebrand), in Rome. This was partly about who should appoint bishops, the Investiture Controversy. Henry declared the Pope deposed, something his father had succeeded in doing to a previous pope, but Hildebrand did not accept his deposition and retaliated by declaring Henry deposed. Not everybody in Germany sided with Henry, and those who did not were called ultramontanists - their loyalty was to a foreigner (beyond the mountains) rather than to their own country. That was the insinuation.
The word came to be used at various periods of history to refer to a strong belief in, or respect for, Papal authority, often beyond that actually claimed or exercised by the pope himself, and often in contrast to local or national authorities, or to dispersed authority. It is typically, but not always, used in a somewhat derogatory sense. This can be seen in the Catholic Encyclopaedia article, written in the years before the First World War, which, on the one hand, asserts that all Roman Catholics are ultramontanists, in the sense they accept Papal authority, yet goes on to say that the adversaries of Catholicism accuse them of ultramontanism but are in reality anti-Catholic even if they claim to be only anti-ultramontanist.
The Reformation depended on anti-ultramontanist principles, that, as Church of England Article 37 says, the Bishop of Rome has no jurisdiction in this realm. The ultramontanist position would be that he should have at least some level of jurisdiction.
During the so-called culture wars, the Kulturkampf, in nineteenth century Germany, the Roman Catholic position was accused of being ultramontanist. This was particularly following German Unification in 1870 which brought many Catholic states under the control of Protestant and somewhat liberal Prussia. Schools were brought under government control and were told to teach what the government and Kaiser said, not what the Church and Pope said. Prussia was no longer officially Protestant, as it had been up to 1848, and as England is, but this had made little difference in practice to Protestant domination, and Prussia, thanks to the exclusion of Austria, was dominant in the new German Empire.
Within Roman Catholicism the word is sometimes used in contrast to Conciliarism. Conciliarists believed that General Councils were of higher authority than the Pope, those who disagreed were called Ultramontanist and their position was the one endorsed at the First Vatican Council, which asserted papal infallibility.
Ultramontanism is not necessarily a consistent set of beliefs on all matters, nor is it a movement or grouping. It is a tendency to support, defend or extend papal authority, but a person might be ultramontanist in regard to doctrine but not administration, or to spiritual but not temporal affairs, or in everything. It is very context-dependent.
Those who opposed the Unification of Italy, preferring that the Pope retain authority in the Papal States, were called ultramontanists, as were those who subsequently argued for the creation of the Vatican State.
The Catholic Encyclopaedia says
It is altogether false to attribute to the Church either political aims of temporal dominion among the nations or the pretence that the pope can at his own pleasure depose sovereigns that the Catholic must, even in purely civil matters, subordinate his obedience towards his own sovereign to that which he owes to the pope, that the true fatherland of the Catholic is Rome, and so forth. These are either pure inventions or malicious travesties.
Here the Encyclopaedia is distancing the Church from the more extreme versions of ultramontanism. It also quotes a Catholic Professor as saying ultramontanists substitute the pope for the church, believe the power of the keys (the pope's authority) should include temporal jurisdiction, that religious conviction can be imposed by force and that the plain teaching of one's own conscience should be sacrificed to external authority. Again this is an extreme form, not one necessarily endorsed by the Church.
Opposition to the freedom exercised by some national or regional bishops conferences, by those who feel everyone should follow directly the pope's line on every issue may be called ultramontanist. (This would be an example of being more ultramontanist than the Pope, who of course authorises the conferences.) Those siding with the Roman Catholic Church against the State, or in opposition to pluralist, liberal, Protestant or other agendas, in whatever place or time, may be called ultramontanist.
In OPs quote
But now that Pope Francis has opened up new spaces, liberals, too, have to learn how to be critically obedient without giving in to the neo-Ultramontanist silencing of those who are disobedient...
the idea that people who disobey the Pope should be "silenced" is described as neo-Ultramontanist. This means either modern day Ultramontanist or extreme ultramontanist. An Ultramontanist might feel that nobody should be allowed to disagree with the pope, perhaps even if the pope has called for open discussion. The writer is alleging ultramontanism, and probably means it in a derogatory way, as implying excessive and undue submission.
Pope Boniface VIII is 1302 said
We declare, say , define, and pronounce that it is absolutely necessary for the salvation of every human creature to be subject to the Roman Pontiff.
This is rather an ultramontanist position to take.
The Latin word mundus means the world, mankind, the universe. Though it has no connection with the word ultramontanist it works as a pun in the sense that an extreme ultramontanist position would be that the pope should be over and above the whole world.
So, although in general we can define ultramontanism as a very high view of Papal authority, its meaning in any given case can only really be understood in context.