- What is exactly does this mean? I assume it isn't just a euphemism for a painless death
- Were there others who also "merited" this?
As pointed out in the comments, the Talmud (Berachot 8A) says that there are 903 types of death in the world, and the least painful one is a kiss. From the way Rashi explains it, it appears that the pain refers to the separating of the soul from the body.
The Talmud explains that the Kiss of Death is likened to removing a hair from milk. This does not require any forceful separation at all.
The Talmud (Baba Batra 17A) (and Rashi) explain that "The Kiss of Death" means that G-d Himself took their soul, not the Angel of Death. The Talmud their says that 6 were not killed by the Angel of Death, but rather G-d Himself took their souls. They are:
The Rambam at the end of Volume 3, Chapter 51 of his Guide to the Perplexed, explains that this form of death is when the prophets meditate so much about the greatness of G-d, and their desire to cleave to G-d is so strong that their soul leaves their body.
He says all prophets had a lower form of this, but Moshe, Aharon, and Miriam experienced the epitome of this, which is why the Talmud singles them out. (I'm assuming he doesn't mention the Avot, because he's focusing on the verse "By the mouth of G-d" which is where the Talmud learns that Moshe, Aharon, and Miriam passed away in this manner.)
in Devarim 32:50, G-d told Moshe that he would die as Aron, right? I found this
Just as your brother Aaron died: [God said to Moses:] “Die with the same death that you witnessed and longed for, that Moses removed Aaron’s first [i.e., upper] garment and dressed Eleazar [Aaron’s son] with it. Then, Moses did the same with the second garment, and the third. Aaron then saw his son in his glory [of his new office]. Moses then said to him: ”Aaron my brother! Go up onto the couch,“ and Aaron went up. ”Stretch out your hands,“ and Aaron stretched [out his hands]. ”Stretch out your legs,“ and Aaron stretched [out his legs]. ”Close your eyes,“ and Aaron closed [his eyes]; ”Close your mouth,“ and Aaron closed [his mouth], and he passed away. Thereupon, Moses said, ”Fortunate is the one who dies a death like this!" - [Sifrei 32:49]
So, would not the "kiss of death" be another way of saying "according to G-d's word", and not like as a literal "kiss"?
In kabbalah there is a very specific technical explanation of מיתה בנשיקה (literally "death via kiss"). See Emek HaMelech section "Gate of Tohu", Chapter 67. Hebrewbooks link - around the second column.
One of the principal spiritual workings of creation is via יחודים (unions). One specific union is נשיקין (kissing), in turn there is a "higher" and "lower" form of this union. The higher refers to the union of the sfirot of Chochma and Bina, and is accomplished by the recital of the Shema. The lower union is via reciting Baruch Shem after the shema.
The principal of מסירת נפש (giving over one's soul) during the recital of the shema is the same as giving ones soul over during death i.e. ideally a person should have the proper intention of creating that union both during recital of the shema and even during death - and that would be "death by kiss". See there at great length where he explains how Moshe's death by kiss was via the union of Chochma and Bina אהיה and יהוה forming the name יאההויהה, yet it is limited to an outward facet as hinted at "kiss" (the mouth is for speech, an externally-oriented organ, for Moshe was on the level of פה אל פה אדבר בו - "mouth to mouth I speak with him").
The Emek HaMelech has lengthy discussions about the non-kiss type of death, i.e. death by קידוש השם (sanctification of G-ds Name or martyrdom, which is a painful death). This type of death is that of the 10 martyrs and in particular Rabi Akiva who longed for that type of death, in order to manifest the more essential union (גופא בגופא ורוחא ברוחא). Indeed he merited this, as he ended up dying while saying shema in the midst of torture.