L-Theanine is an amino acid that is not common in the diet and not one of the essential amino acids or even one of the common nonessential amino acids. It is found in meaningful amounts in tea, but otherwise supplementation is the main source of it. L-Theanine has structural similarity to glutamine and both neurotransmitters that are produced from it (GABA and glutamate). It is known to reach the brain and influence its function following oral ingestion.
A reduction in anxiety, stress, and greater ability to relax is its most likely effect, and while still somewhat preliminary, studies suggest a benefit for these purposes. This effect likely also leads to an increase in sleep quality due to deeper sleep, but it's not apparent whether or not it's effective for improving insomnia in general.
Other than that, the benefits are more murky, and while potential improvements in attention and executive function have been noted, but much more research is needed before having much confidence in that, and other cognitive benefits are dubious.
It may have a mild antagonistic effect on caffeine and reduce the high blood pressure and jitters, taking "the edge off", but more research is needed to confirm this.
Interestingly, the relaxing and attention promoting properties of L-theanine coupled with the lack of sedation may have its most significant supplemental role in attenuating the 'edge' of many stimulants. A combination of theanine with caffeine (200mg each) is noted to be synergistic in promoting cognition and attention.
Very scarce research has suggested a potential reduction in alertness, which makes sense due to its calming effects. However, this research needs to be replicated and explored more deeply in more activity contexts.