Explain why Elizabethans were worried about vagrants in 1558–1588.

Poverty was a very serious problem in Elizabethan England. But the existence of large numbers of destitute men and women with nowhere to live—vagrants—was, if anything, even more serious. Far from seeing vagrants as innocent victims of forces beyond their control, the authorities regarded them with suspicion, seeing them as potential rebels.
Strange as it may seem, the government was genuinely worried that vagrants would form themselves into gangs and wander around from place to place, undermining the social order as they went. The government was run by, and largely for the benefit of, men of property, and these men were concerned about the threat to their property posed by those with nothing to lose.
It was in response to these concerns that Parliament passed the Poor Laws, which gave local magistrates extensive powers to provide financial help for the "deserving" poor, i.e. those who couldn't work due to age or disability, while cracking down on so-called sturdy beggars: those whose poverty was put down to outright laziness.
The punishments for beggars were harsh in the extreme. The Vagabonds Act of 1572 stipulated that wandering beggars could be whipped, bored through the ear, or even put to death if they were caught begging. Under a later piece of Elizabethan legislation, the Poor Law of 1601, begging was banned outright. Anyone caught begging would be whipped and sent back to their place of birth.