This is a really interesting question, and one that has been raised by scholars in this field a number of times. The general consensus still tends to fall to agreement that yes, Shakespeare's female roles were indeed played by young boys, usually about 14-16 years of age.

 

I want to address the last part of your question first, that "I just don't understand how a 15 year old could have played Lady Macbeth or Queen Gertrude and be taken seriously", because this is a fair enough query, but really is what lies at the root of what you're asking. Perhaps you don't seen how a boy actor playing Lady Macbeth could be taken seriously, but unlike the early modern playgoers at early performances of Macbeth, you have no cultural conditioning either to expect this, or take it seriously. In fact, early modern playgoers were so accustomed to boy players that some of the most popular players companies in Shakespeare's day were children's companies composed entirely of young boys. Christopher Marlowe and Thomas Nashe's play, Dido, Queen of Carthage, was written for a boys' company and was played by the 'Children of the Chapel' according to the title page of the 1594 quarto printing of the text. Not only does the play demand a number of female characters, but the title character is of course a woman, with a substantial number of lines. Based on Virgil's Aeneid (and indeed, including a fair bit of racy material) early modern audiences clearly took no issue with children performing fairly serious, weighty material, including roles which called for mature, articulate women (the Penguin edition of Marlowe's Complete Plays has some interesting content on the use of boy players in the notes section, if you are interested).

 

In terms of Shakespeare's use of boy players, one of the more interesting pieces on this that I have seen is Stanley Well's article, 'Boys Should be Girls: Shakespeare’s Female Roles and the Boy Players' (New Theatre Quarterly, Volume 25, Issue 2 May 2009 , pp. 172-177). Wells argues that Shakespeare's plays tend to call for a remarkably stable number of boy players across his oeuvre - around 4 - which would seem to indicate that there were around 4 boys available in the actors company at any given time, and Shakespeare's plays evidence 'Shakespeare’s professional skill in cutting his coat according to his cloth' (p. 177).

 

Some written evidence exists detailing audience's responses to female roles. For example, Gary Taylor's chapter in The Cambridge Companion to Shakespeare on Stage (titled, 'Shakespeare Plays on Renaissance Stages') includes the commentary of an Oxford playgoer who saw Othello in 1610, praising the actor who played Desdemona and the fact that she 'acted her whole part supremely well' (p.10).

 

To conclude this section of my answer - believability is in the eye of the beholder, and with no women permitted to act on the stage in this period, early modern audiences were quite happy to accept young boys in women's roles.

 

Believability aside, most of the other criticisms I have seen of the suggestion that boys played female roles tend to focus on the implausibility of such young boys memorising and performing substantial chunks of text. Even among those who are happy to subscribe to the idea that Shakespeare used boy players, there are those who question their stamina or abilities as actors. If you read on to the next section of 'Shakespeare Plays on Renaissance Stages' (which is available on google books) you'll see that Taylor moves on to discuss the brevity of female parts in Shakespeare, suggesting that this is because the boy actors couldn't be taxed with more lines to remember. Others suggest that this may be more as a consideration to the boys' young, and weaker voices. An actor who has lost their voice from vocal strain is of very little use, after all!

 

There is no question that many of Shakespeare's female parts are shorter than those of men, but I do want to end by offering a bit of a challenge to Taylor's assertion that the boys struggled to remember substantial parts, because he is by no means the only critic to espouse this view. Plays written for children's companies certainly make no bones about having young lads remember huge swathes of text (additionally, other playwrights for mixed players are sometimes equally generous with lines for boy players), and this capacity for memorisation has study historical evidence. The humanist education in this period focused on memorising written material as one of the figures of rhetoric, which Hester Lees-Jeffries describes as 'integral to all education, how to read, write, think and organise one's ideas' (Shakespeare and Memory, Oxford: OUP, 2013, p.16). Many children's companies grew out of choir-school companies, and these boys would have been provided an education as part of their membership in that company (see Andrew Gurr, The Shakespearean Stage 1574–1642, Cambridge: CUP, 2009). We can therefore expect that they were trained in memorising substantial chunks of material as part of that education, and would have had no trouble with memorising parts for a play.

 

The skills required of boy actors is an increasingly popular area of research these days - I have seen amazing research turn up in conferences a few times in the last year. By all accounts, these kids seem to have been really, really talented actors with physically demanding roles.

 

I hope that's cleared things up somewhat, feel free to let me know if you have any other questions.