Why do kids shows have so many musical numbers

Depending on the age of the children being targeted there could be a few reasons...

Babies & Toddlers

For younger children various studies have shown that listening to music can help their development.

For example, singing songs teaches children about language such as how it is put together. When singing, words and phrases are slower and are easier to understand. It will help them to build up a vocabulary of sounds and words before they can even understand the meaning.

Listening to music also helps to put babies and toddlers into a relaxed and receptive state.

Most babies tend to have an affinity to music as well, they will sway or move with the rhythm which will help build coordination and motor skills.

As a lot of TV for this age range is aimed at helping them learn words and develop, using music as a tool towards this end just makes sense.

Older Children

For older children who are making the progression from the baby & toddler music and singing oriented TV, to TV aimed at their own age. The music might be there to help ease the progression from fun learning to TV that contains more serious dialogue.

There is also the cynical aspect that many of the shows on Disney and Nickelodeon contain actors that are also singers and they will obviously try to push the singing aspects of these to provide an additional revenue stream, and to help grow that performer as a musical artist and help generate song sales, possibly long after the TV show has finished.

Examples of these would be

• Bella Thorne - Shake it Up
• Joe Jonas - Camp Rock
• Zendaya - Shake it Up
• Nick Jonas - Camp Rock
• Demi Lovato - Sonny with a Chance
• Selena Gomez - Wizards of Waverly Place
• Ariana Grande - Victorious
• Miley Cyrus - Hannah Montana

And both the lead stars in your comment Example "Austin & Ally", Ross Lynch and Laura Marano have released albums - albeit soundtrack albums for Disney programmes & films. Ross Lynch is also in the Pop-Punk band R5.

Adults

Adults have grown out of the need to develop language in this way, as other methods are likely to be more effective.

Adults are also probably less likely to fall for the marketing technique for the singers, after-all that is what X-Factor and the like are for.

So adults that watch films or TV with musical numbers in do it purely for the pleasure. Thus there is a specific musical genre and TV producers do not feel the need to place it into everyday programming.

It's a relatively modern trend. In the 19th century, a night at the opera was the thing adults would do; and in fact, in some cases, a visit to the theatre actually strayed far more into the other meaning of "adult entertainment" (actresses were not considered the most virtuous of persons). There was a progressive shift toward making musicals more family-friendly, eg by Gilbert & Sullivan, and as the 20th century began, musicals were something that everyone enjoyed - as were cartoons. Today, you're considered mildly weird if you genuinely enjoy something like "The Sound of Music", but on the other hand, there are always those little nods to the adults (TVTropes has an entire section for "Parental Bonus"), as it's well-established that everyone should be able to enjoy this kind of performance.

My suspicion is that it's just too costly to find an entire cast who can sing, act, dance, AND take direction. Filmmakers have to choose how much to spend on the cast, how much to spend on the music (songs vs BGM), how much to spend on SFX, etc, etc, etc. The budget has to be squeezed somewhere.