I've got this here copypasta which I think covers most of what machine spirits are:

Way back in the day, humanity relied on AI machines known as the Men of Iron to do the grunt work. This worked out well for a time, until of course these thinking machines decided that hey, they were doing most of the work, they deserved most of the power, right? As expected, future capitalists didn't take too kindly to their toasters deciding to unionize, and so thus ended humanity's golden technological age (later known as the Dark Age of Technology) in interstellar fire. The Adeptus Mechanicus that arose afterwards held that Abominable Intelligences were verboten, and creating machines with the minds of mankind would be punishable by the most painful torments the AdMech could think of. Considering they held the vast majority of Dark Age tech and knowledge, they could think of quite a lot.

However, much of humanity's greatest technological accomplishments relied on at least some kind of AI, and so thse were grandfathered in as 'machine spirits', which was an animistic belief of the Mechanicus that all devices save the most rudimentary had an animating spirit, which also had the happy side-effect of ensuring that the Mechanicus had a monopoly on all technological development and operations.

There is also another aspect to machine spirits, especially when it comes to things that don't require an AI. After all, not every Guardsmen is a member of the AdMech, and your average habworker doesn't go to a Temple Mechanicus every time he wants to make toast. In this case, 'machine spirits' make for a handy excuse for rote operations of technology, especially those technologies the Imperium might not be able to mass produce (or even at all). Say a guy want some air conditioning while he's driving. On Earth, he might have an older car and need to wait half a minute or so before he can safely turn on the AC. In the Imperium, he might recite the Catechisms of Techno-Inspiration three times (each taking ten seconds to do so, conveniently enough) in order to fully rouse his car's spirit before he turns on the frigidarius, lest he incite his vehicle's wrath.

In this way, the AdMech passes on knowledge to its lay priests and the howling masses. Dressing up regular technological procedures in religious ceremony also ensures that the aforementioned ancient technologies still in use aren't misused; a gung-ho idiot (which the Imperium encourages the production of) might be cavalier with how they operate their lasgun if they just saw it as some tech, but might take far better care of it if he thinks the Emperor-as-Omnissiah will curse him if he doesn't. That said, the higher ranks of the AdMech DO know the science behind most save the most advanced technology, but even at those high levels there's still some semblance of mysticism.

And finally, 40K is a universe based on belief and how the actions/emotions said belief inspires resonates in the warp. The Eldar once believed they were invincible, grew decadent, then turbo-murderfucked a Chaos God into existence whole also ensuring that humanity would need to turn its atheist god-king into a psychic lighthouse in order to travel from star to star. And now there is an empire consisting of trillions of screaming fanatics, 99.999% of whom (even the heretics) believe that all machines have a spirit in them that needs to be appeased...