There are different ways of understanding how Reiki works, and no single explanation has been universally accepted.
The perspective below reflects my own understanding of Reiki and is heavily influenced by the work of James L. Oschman, particularly his book 'Energy Medicine: The Scientific Basis'.
There have always been a very restricted number of people that, through many years of meditation or prolonged fasts, have been able to develop a deep connection to the universal energy (for example Shaolin monks).
Mikao Usui's discovery is that this connection to the universal energy can be obtained in a simpler way going through an initiation that he invented and it's composed of 'attunements' in which this capability is passed in person from teacher to practitioner.
To my knowledge the only aspect that has been possible to scientifically verify is that, following this process, the practitioner's hands do acquire the capability of displaying more energy (assessed through Kirlian photography or other instruments).
My current understanding is that Reiki involves resonance or entrainment between practitioner and client more than a direct transfer of energy. In this view, the practitioner's state helps encourage a similar state in the client.
During a Reiki session the brain of the practitioner (in particular the pineal gland) synchronizes with the Schumann resonance created by the Earth's magnetic field, and goes into a alpha and theta brain waves mode (the ones you generally achieve through meditation), these waves travel through the practitioner's body and arrive to their hands; instead of weakening they actually gather strength as they travel to their hands .
The client's body parts in contact with the practitioner's hands start to resonate with these frequencies, like when you put 2 tuning forks next to each other. This stimulates the sympathetic nervous system, which in turn signals the body to start a recovery process toward well-being.
This interpretation is taken from James L. Oschman's book 'Energy medicine, the scientific basis'.
These are a few links where this process is explained in more detail:
https://www.reiki.org/articles/science-measures-human-energy-field
https://www.balanceisqi.com/post/reiki-connecting-you-to-alpha-and-theta-brainwaves-of-healing
https://www.poeticmind.co.uk/wellbeing/the-science-behind-reiki-what-happens-in-a-treatment/
https://serenityattunement.com/the-science-behind-reiki/
This is a study that shows the effects of Reiki:
'Use of Reiki as a biofield therapy: An adjunct to conventional medical care'
https://oatext.com/Use-of-Reiki-as-a-biofield-therapy-An-adjunct-to-conventional-medical-care.php
This is an extract from the 'Conclusion' of the study:
'There is enough research evidence to support that Reiki facilitates relaxation, reduces pain and anxiety, promotes healing, improves fatigue and depression, and enhances overall wellbeing. Although there has been an increased interest among researchers to examine the efficacy of Reiki therapy, more well-designed and controlled studies are needed for it to be accepted within the mainstream health care system. Because of the simplicity of this modality, the author will not be surprised if in the near future, the mainstream health care professionals begin advising the use of Reiki along with a balanced diet and exercise for prevention and treatment of illnesses.'
This is a study that shows that Reiki is better than placebo:
'Reiki Is Better Than Placebo and Has Broad Potential as a Complementary Health Therapy'
https://pmc.ncbi.nlm.nih.gov/articles/PMC5871310/
This is an extract from the 'Discussion' of the study:
'Viewed collectively, these studies provide reasonably strong support for Reiki being more effective than placebo'
Regarding the benefits you can expect from Reiki, please refer to this link:
https://www.webmd.com/pain-management/reiki-overview