Link to the document (which is also linked in the article).

The article has a couple of inaccuracies and unjustified assumptions:

First of all, there is no Men's category in chess, only Open and Women's. The Open category is, as indicated by its name, open to everyone, men, women, cis, trans and anyone else, and it is not unusual that women compete in it (the most prominent case was Judit Polgar). This new FIDE document doesn't change that, so a trans person is allowed in competitive play, contrary to what the title suggests. The Women's category is restricted to only women, and this document explains how these restrictions are applied to trans people. Note that the existence of a Women's category is controversial on its own.

Similarly, there are not Men titles and Women titles, only General titles (Grand Master, International Master,...) and Women titles (Women Grand Master, Women International Master,...), with the Women titles having lower requirements (again, very controversial). All the top women players are GMs, not WGMs. The FIDE document states that trans men will have their women titles abolished (unless they transition back), but they retain their general titles, and in case they don't have one, their women title will be transferred to the general title of the equivalent level. The article incorrectly states that trans men "are set to be stripped of any titles if they were won pre-transition".

Now, to the meat of the topic: the eligibility of trans women for the Women category:

Article 2.3 states that any person who has legally transitioned can certificate so to their National Rating Officer to change their gender in FIDE's database. Once that that change in the database is done, trans women are restricted from Women events until a decision is made by FIDE Council "at the earliest possible time". In these cases where the transition is backed up by legal documents, the decision by FIDE should be a formality and take little time.

Of course, this is only possible for countries who recognize gender changes. In case the National Rating Officer rejects the change, article 2.4 allows an appellation to the FIDE QC, and after that to the FIDE Council. In these cases the decision by the FIDE Council may take up more time, and FIDE sets a deadline of two years for that decision (but it doesn't mean that every case will take two years).

And finally, just to be clear, all of this only applies to trans players who need to change their gender in FIDE's database. Trans players that always have played under their post-transition gender, or that have changed their gender before this regulation, don't need to go through this process. This is the case, for example, of Yosha Iglesias, the player mentioned in the article, and that is listed as female in FIDE's database, and consequently she can play in any Women's event she wants.

All in all, I think there is much unnecessary red tape, especially for the trans people more at risk, those who live in countries where they are persecuted. And that the excessive red tape is most likely based on transphobia. But the situation is not as extreme as many other sports that have directly banned the participation of trans players.