Functional medicine is not a form of medicine supported by mainstream medical authorities. Most practitioners are not required to join any form of professional organisation that would uphold standards and are thus not regulated in a compulsory way, so it is important that someone thinking of seeing such a practitioner protects their own interests by ensuring that the practitioner is accredited by the Professional Standards Authority for Health and Social Care (PSA). Usually, these associations or registers demand that practitioners hold certain qualifications, and agree to practise to a certain standard.
As a general rule these practices can be useful in addition to the therapy offered by your GP or other NHS doctor, indeed there are NHS GP's and other clinicians (eg nurses) who are also practising functional medicine and other forms of alternative therapy. They should never be used instead of the medication prescribed by your GP.
Functional medicine concentrates on the impact the environment has on our health which covers a lot of possibilities, often in chronic illness that cannot be cured by conventional medicine. One reason for its popularity may well be because conventional doctors tend to be short on time and short on effective treatments for some chronic diseases. Functional medicine is less restricted in scope and may be able to devote more time to each patient.
Functional and mainstream medicine can complement each other. For example we know that diet is an important factor in our health for several reasons, and science is beginning to tell us that the microbial contents of our gut has a profound impact on our health too. It appears in early experiments that a diet rich in fibre supports a richer range of microbes in our gut (our microbiome) and that has a positive impact on our health. Conventional doctors are not yet able to offer advice and treatment based on supporting our microbiome and wont until experiments and clinical trials prove it is the right thing to do - there are nearly 1000 clinical trials currently ongoing. However it has been common practice for many years to advise on a good diet, which already includes plenty of fibre.
Functional doctors feel less restricted with their advice and they are already proposing that we can modify our mental health and mood by adjusting our diets to accommodate a healthy microbiome. A GP that is treating depression might offer a prescription for antidepressants or cognitive behavioural therapy whereas a functional doctor might also suggest non-drug helpful options such as changes to your diet. Arguably each approach has its strengths and when they are used to complement each other like this there could well be some benefit to the patient.