I've heard plenty of Beatles songs and they're fine but they don't really blow my mind and I don't understand the reverence with which they are viewed today. What am I missing?
I definitely have ideas to present on this. I wanted to mention up front that I lived through the Beatles era (I'm almost 65), so my thoughts will definitely be different from those who learned about the group after their breakup in 1970.
I will break down their importance in three ways: As live performers, as songwriters, and as producers of records.
Of the three, their role as live performers was likely the least important to the average person. While they toured extensively until August, 1966, most of their concerts beginning with their American appearances in early 1964 were short (usually not much more than 30 minutes or so) and were usually punctuated by constant screaming, meaning their performance abilities were somewhat muted. However, the group played extensively in London and Europe for years before they first started recording in late 1962. There are a number of bootleg recordings from this early period that showed them as a tight-knit rock and roll band who learned a great deal of how to work together. They performed a large number of covers of usually American rock & rollers, but even during this period were performing their own songs. There are also great recordings they did for the BBC that help to provide their musicianship.
So how were they different from other rock & roll musicians of the period? From the beginning they included tight harmonies, often from three of them at a time (John, Paul, George). This showed them performing as a group, rather than the concept of solo artists or solo artists with a backup group, which was more prevalent in the period. They also traded off as solo singers and this gave them a more diverse sound. In the end, it was their ability to perform live as a group that led to an eventual record signing.
Their second major role was as songwriters. While their entire recorded content consists of not much more than 10 hours total, nearly all of the albums were self-written material. In the 1950s and early 1960s, most groups and artists performed material written by others, which can be seen as a less important skill. While the group included cover songs on most of the first five albums, they still featured their own material more and didn't release any singles of music written by someone else, except for the occasional B-side or records released in Canada or some other locations. Some of their early fights with the record company came about because of their desire to only do their own songs. Obviously, this was unheard of.
One of the other important aspects of their writing was how they began to experiment with the song form, which was pretty much set in stone at the time (verse/verse/chorus/verse/chorus/middle eight bars/chorus, etc.). They also began to make sure that their lyrics were more in-depth even in their early period. They were recognized early on for how they wrote. While most of the songs were listed as written by Lennon/McCartney, it's pretty much set in stone that whoever sang the song was usually the main writer. Lennon very quickly developed lyrics that often showed unhappiness or dissatisfaction, with songs like No Reply, Help, and others clearly providing thought-provoking songs. McCartney was more of a traditional love song artist, but both produced an incredible amount of hit music. They were able to prove that musicians could write better music than songs written by others. This would become the norm going forward (and more interestingly, didn't largely change in the country/western format). To this day, their writing catalog is incredibly valuable for the number of songs performed by others.
So that brings us to the band members as recording artists. In August, 1966 the group performed their last regular live concert, although it wasn't advertised as such at the time. They were tired of the grind of the road, particularly performing where nobody could really hear them anyway! In addition, they were writing more complex material that couldn't be as easily performed by a traditional 4-member rock and roll group. Therefore, beginning with the Revolver album, they became a studio group only.
Even before this, though, their recordings were getting more complex, with additional instruments, double-voice tracking and use of overdubbing and mixing to produce a good sound. If you listen to early rock and roll, the vocals tended to be more important than anything else and most songs were not recorded in an complex way. By even the second album (With the Beatles), the group began adding piano and other instrumentation to make their songs more interesting. They were luckily paired with George Martin as their producer, and he went along first as a teacher and then as someone who facilitated what they wanted to do. There are extant snippets of some of the early recordings that showed their willingness to perform songs multiple times or even just small sections to try to put together the best recorded versions. Clearly, they decided that their recordings were just as important, if not more important than anything they did live.
One of the best ways to get a good sense of their skills is to start at the beginning and go from album to album to see the way they grew. Once they hit Revolver, when they didn't care at all about performing the songs in a concert, they really began to experiment and hit their real importance. While they largely stuck to the traditional song format they had always used, they just refused to be normal. Songs like In My Life, Penny Lane, Strawberry Fields Forever were constructed normally, but featured instrumentation and chord structures that were completely different from anything else before. Lennon in particular began to play more with language in his songs, continuing to clearly present the lyrics as being as important as the music. Listening to a song like A Day in the Life, it's hard to understate their talent.
All in all, they developed the idea of selling records as being an important part of their output. Other groups would begin to understand this, but The Beatles worked hard on every aspect of each song and each album. It was always a major event when a new single or album came out.
So, going back to your initial question: The true talent of the Beatles can best be examined in a chronological fashion, comparing what they did to similar material released by others at the time, as well as how they affected and influenced virtually every other artist from the period and even after they broke up. The early albums presented them as a great pop group. The later albums showed them as great musicians.
I know I went long, but I hope that the detail assists in your questions!
I've answered a bunch of questions along these lines, including:
What exactly was the influence of The Beatles on the long-term?
How big is the importance of album The Beatles - Sgt. Pepper's Lonely Hearts Club Band for music?
Hope that helps!