The essential question that Shakespeare explored in his plays is, "what does it mean to be a human being?" The genius of Shakespeare is that he manged to show us ourselves in every conceivable light. It really doesn't matter when the plays were written since they are about the human condition which is timeless.
Shakespeare was an incredible observer of his fellow human beings. We, as human beings, may dress differently or have more technology than people in Shakespeare's day but we are still motivated by the same desires and have the same feelings.
The themes of his plays are as universal as his characters. For example, in his history plays, he explores the question of power and how to govern. His examples show men who govern poorly (Richard II), who take power (Henry IV parts 1 & 2), who who don't desire power but live up to the challenge (Henry V), who mean well but are ineffective (Henry VI parts 1, 2 & 3), and who becomes a tyrant (Richard III). Each of these lessons, he draws from English history but can be applied throughout history right up to the present day. We still good leaders, bad leaders, ineffective leaders, and tyrants. Macbeth is another example of a tyrant who takes power by forces. All we have to do is read a daily newspaper to see the contemporary nature of Shakespeare.
The relationships between men and women in their various relationships are also explored. A feud and forbidden love and how it affects a society is explored in Romeo and Juliet. Today we have racial and religious differences which can divide a society. Teenagers identify with the characters in this play which is why it is taught in high school.
Parental relationships are also explored in his plays as well as family relationships. There are numerous examples, particularly in the Late aka the Romance plays. But the comedies also show us sibling relationships, like Kate and Bianca in The Taming of the Shrew or Don Pedro and his brother Don John in Much Ado About Nothing.
These are only a handfull of examples but it is easy to see that Shakespeare managed to capture mankind in all his glory and all his shame and everything in between. This is the reason that he is for "all time".