In the Star Trek universe the Federation seems to encompass a lot more (habitable) systems than the Romulan and Klingon empires relative to its size. Was this ever explained in the show?
One possible explanation is that the Federation is much more diverse in regard to member species and the the rights (for colonization) that these members have. That way the range of planets that are inhabitable is much wider.
The Klingon Empire, the Romulan Star Empire and the Cardasian Union are practically single-species empires. Any other species are second-class citizens at best and slaves at worst.
So the Federation has following advantages when it comes to colonization:
- Citizens are free to colonize planets without species restrictions
- Some planets that would be not appropriate to the "main" species (humans?) can still be colonized. For example a planet that has average temperature of 0 °C will be still OK for some species that evolved on such a planet. Humans on the other side will feel quite uncomfortable there.
- (as noted by Richard) When a new member joins the Federation, they usually have already done exploration of the area around them and they contribute this knowledge. The other empires expand by either aggressive colonization or conquest. In both there is none or little exploration data to be collected and much less people from the conquered species are ready to cooperate in colonization efforts.In the example with Bajor - the Cardasians had to fight constantly with the underground resistance and this consumed their resources - resources that could have been spent on colonization.
Who created the map (in-universe)? Note that it is labelled "United Federation of Planets".
In reality, maps tend to be creator-centric, with many biases towards their own region. Especially, if the map is for a region, the surrounding area may be shown in less detail to make it easier to see the detail in the "important" region.
The Federation is inclusive by its very nature. It's friendly. It wants people to join the club. That in itself goes a long way towards explaining its huge territory.
By contrast, the Romulans are isolationists and the Klingons expand via conquest, which is a lot harder than smiling at someone and inviting them to come over for peace and biscuits.
As for the planet density within Federation space, that's almost certainly down to audience bias and intelligence considerations. In how much detail do you suspect the TNG-era Federation knows the "geography" of the Romulan Star Empire? And why would the TNG-era Federation internally distribute well-detailed maps of Klingon space? Nobody's going to be hopping on a transport there to get to work each day.