An overseer, then, must be above reproach, the husband of one wife, temperate, prudent, respectable, hospitable, able to teach…
1 Timothy 3:2

… but hospitable, loving what is good, sensible, just, devout, self-controlled, …
Titus 1:8
(emphases mine)

Not only is being hospitable essential for elders, it is also a requirement for all believers:

Be hospitable to one another without complaint.
1 Peter 4:9

This is a compound word that is specifically a love of strangers. Hospitality flows from a supreme love for God that results in a biblical love for people – the fulfillment of the second great commandment.

And He said to him, “‘YOU SHALL LOVE THE LORD YOUR GOD WITH ALL YOUR HEART, AND WITH ALL YOUR SOUL, AND WITH ALL YOUR MIND.’ This is the great and foremost commandment. The second is like it, ‘YOU SHALL LOVE YOUR NEIGHBOR AS YOURSELF.’”
Matthew 22:37–39

In New Testament times, traveling believers often had to rely on fellow believers for accommodations. Travel could be dangerous and Christians were not always well-received. The book of Hebrews encourages us to show such kindness to strangers.

Do not neglect to show hospitality to strangers, for by this some have entertained angels without knowing it.
Hebrews 13:2

Hospitality is a way for elders to get to know those who are new to their congregation. It is also a way for new people to see the life and testimony of the church elders. This is a reminder that biblical leadership must never be isolated to meetings that are held behind closed doors. Elders are fully engaged in the lives of the believers they serve.