What problems arise from having guns in a home?

The primary problems are accidents, suicides and homicides.

Improperly stored guns (i.e. those which are NOT locked in a secure gun safe, and which are not properly unloaded and/or have ammunition stored separately) are a major hazard to children. Children are naturally curious, and, even with proper training and education, are much more prone to making poor decisions around available firearms. Unfortunately, the majority of firearms are NOT properly stored in the US - most lack at least one of the preceding criteria, and thus, are accessible to children.

Mishandling of firearms by adults also results in a non-trivial number of accidental shootings, usually only a few fatal, but most serious (requiring hospitalization) nonetheless.

Firearms are also the leading method of suicides, both attempted and successful. Suicide attempts in homes with a firearm are notably higher than in homes with no guns, and successfulsuicides in gun-available homes are radically higher. The main issue is that a gun is a fast, simple way to kill oneself; other methods require significantly more effort and planning, and the majority of suicides are impulse decisions. Not to mention that a firearm is far, far more likely to result in death than any other method used by a suicidal person. For example, the number two method of suicide is via poisoning, and the success rate from poisoning is roughly one-quarter that of firearms.

Finally, the use of firearms in domestic disputes and domestic violence is extremely problematic. Having a firearm available in the house means that any domestic dispute or abuse is radically more likely to result in some sort of shooting, and the vast majority of those shooting are not defensive one (i.e. it's not the victim shooting the aggressor/abuser, but the opposite).