What does je puis mean?

Je puis is actually incorrect. The correct form is:

Je peux. -- I can (the verb is an irregular from of pouvoir).

That said, if you form an inverted question, for example: "Can I help you?" you would say:

Puis-je vous aider? -- Can I help you? You might think it ought to be peux-je and not puis-je but that's where French is throwing yo a curve ball. That's just an exception you have to memorize.

I don't think you can say that "je puis" is incorrect, it's just a bit old-fashioned or formal. For example, you'll find Michel Barnier using this in some Brexit statements:

"Je parle au nom des 27 chefs d'Etat et de gouvernement, si je puis dire aussi au nom du Parlement européen, il y a unanimité du côté de l'UE pour que dans l'accord de retrait que je dois mettre sur la table (...) il y ait un règlement sérieux avec les Britanniques"

Here is what my French go-to native speaker says: "Jamais dans la vie courante. C'est d'un style très littéraire, ou ancien."

Hi Alan,

Je puis is certainly not incorrect and as you pointed out 'si je puis dire' is a very common expression.

I had a look through my electronic copy of "Le petit prince" by Antoine de Saint-Exupéry and every one of the numerous instances of 'I can' is rendered as 'Je puis' and none as 'Je peux'. This novella, voted the most popular book of the 20th century, with a recommended reading age starting from 9 years, can hardly be deemed either supremely literary or archaic.

Or perhaps it could be that these aristos cannot forgo their lofty language even when writing for children. ☺

Je puis is very formal and can sound pompous, you might  hear it said between friends for comical effects.

In some shops, someone might ask you "Puis-je vous aider?" which is the very formal way of adressing customers like,  "Que puis-je faire pour vous?".

In everyday life, you will say: 

Est-ce que je peux vous/t'aider?

Quest-ce que je peux faire pour vous/toi?

Hope this helps!