The bareness of the stage in The Globe and other theatre venues used by the Lord Chamberlain's Men and the King's Men should not be taken to imply that there weren't any other elements of stagecraft. G. Blakemore Evans points out that
the actors depended on relatively lavish costuming, most of it bought second- or third-hand (...), and a probably fairly sparing use of common properties (bed, throne of state, chairs, stools, trestle tables, tree, rock, chariot, etc; (...)).
(G. Blakemore Evans: Elizabethan-Jacobean Drama: A New Mermaid Background Book. London, A & C Black, 1989, page 70.)
Both Shakespeare's plays and plays by Shakespeare's contemporaries suggest that the theatres used certain special properties. For example, thunder could be suggested by "rolling a cannon-ball down a wooden or metal trough, or by drum rolls" (Blakemore Evans, page 70); "rain and hail storms were simulated by 'raining' down dried peas from the 'heavens' (from which a deus ex machina, ..., could be lowered; ...)" (Blakemore Evans, page 71; see also C. Walter Hodges's conjectural reconstruction of The Globe, where the 'heavens' can be seen above the stage). There was also stage blood (e.g. Romeo and Juliet).
To take the examples from the question one by one:
- "Enter PROSPERO above, invisible." Prospero may have appeared on the "balcony" above the "inner stage" (the "inner stage" is behind the curtains in C. Walter Hodges's conjectural reconstruction of The Globe).
- "Enter several strange Shapes, bringing in a banquet". These would be actors wearing unusual costumes carrying in a table or a trestle table on which "food" can be seen; they also carry it away again.
- "Thunder and lightning". The thunder would be made using the means mentioned above (cannon ball or drum rolls). How lightning would be simulated is not clear, especially during performances in public theatres, which would start at 2 or 3 p.m.
- "Enter Ariel, like a harpy, claps his wings upon the table". Ariel, wearing an appropriate costume, might either walk onto the stage, or lowered down from the 'heavens'.
- "with a quaint device the banquet vanishes". This is less clear. (It is important to know that the theatre may have had a trapdoor (see C. Walter Hodges's drawing again), however, it is not clear how this trapdoor may have been used to make the banquet disappear without removing the table. Alternatively, the banquet was actually on a plank on top of the table, and that plank was hauled up to the 'heavens'.)
- "He vanishes in thunder". For the thunder, see above.
For the above list, I have assumed that The Tempest was performed at The Globe or the Blackfriars Theatre. Except for costumes and instruments, some aspects of stagecraft and scenography mentioned above would not have been available during the performance at court on 1 November 1611 (see also The Tempest, edited by Stephen Orgel. Oxford University Press, 1987, page 2).
However, there is another aspect to consider. Ralph Crane was a scribe who is known to have worked for the King's Men. Eugene Giddens writes that "Crane's stage directions often seem like descriptions written for the benefit of readers" (Giddens: How to Read a Shakespearean Play Text. Cambridge University Press, 2011, page 32). The best-known example of this occurs in The Tempest, Act III, scene 3 (the banquet scene cited in the question):
Enter Ariell (like a Harpey) claps his wings upon the Table, and with a quient deuice the Banquet vanishes.
Presumably the acting company would need to be aware of the kind of quaint device that the author had in mind, but this direction seems to describe the action after it has happened, and for a reader's imagination.
(Note that none of this implies that Ralph Crane filled in stage directions where none existed before, as was suggested by the question based on an older answer to a related question.)