The quote expresses Shakespeare's view of human nature--that mankind is basically evil. Specifically, in this quote, the playwright suggests that evil reproduces itself. If someone premeditates any type of crime or "sin," that premeditation causes the crime or sin to spin out of control and to become more than the perpetrator intended. The word "ill" has several interpretations. It is used in place of "evil," but it is also connected to the phrase "ill will" which means that someone intends to harm another or wishes for someone else to experience difficulties.
In the case of Macbeth, neither he nor Lady Macbeth had any idea at the beginning that their "ill" plot would lead to so many deaths or that it would have such a profound psychological effect upon them.
This is the whole quote. In it, Macbeth tells Lady Macbeth, without saying so directly, that he plans to kill Banquo and his son, Fleance. Banquo knows too much about Macbeth and must be eliminated. When he says "Things bad begun make strong themselves by ill" he means this: The murder of King Duncan was but the beginning; more killing will and must take place for Macbeth to live in safety.
Macbeth:Be innocent of the knowledge, dearest chuck,
Till thou applaud the deed. Come, seeling night,
Scarf up the tender eye of pitiful day;
And with thy bloody and invisible hand
Cancel and tear to pieces that great bond
Which keeps me pale! Light thickens; and the crow
Makes wing to the rooky wood:
Good things of day begin to droop and drowse;
While night's black agents to their preys do rouse.
Thou marvell'st at my words: but hold thee still;
Things bad begun make strong themselves by ill.
So, prithee, go with me.
The quote "Things bad begun make strong themselves by ill" is obliquely linked up with the earlier quote "We have scotch'd the snake, not kill'd it". It indicates Macbeth's mental perplexity and perturbation. He says to Lady Macbeth "full of scorpions is my mind". Macbeth is guilt conscious. He knows now that crime once committed does not allow you peace of mind and leads you to the commission of several other crimes for various reasons, it does not spare you but instead compels you to wade through a criminal career till your death.