How could he explain to a non-Muslim the reason why he grows a beard?

Firstly: 

Letting the beard grow and leaving it alone, is obligatory, as has been explained previously in several answers. Please see the answer to question no. 1189 . 

Secondly: 

What you have to do if any non-Muslim person, no matter what his religion is, asks you the reason why you are growing a beard is to explain to him the shar‘i reason why you are doing that, which is that you are only doing it in obedience to Allah and His Messenger, and that the beard is one of the sunan al-fitrah, i.e., actions that are in accordance with sound human nature with which Allah has created His slaves who affirm His Oneness. 

Muslim narrated in his Saheeh (261) that ‘Aa’ishah (may Allah be pleased with her) said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “Ten things are part of the fitrah: trimming the moustache, letting the beard grow, using the siwaak, rinsing the nose with water, cutting the nails, washing the finger joints, plucking the armpit hairs, shaving the pubes and washing the private parts with water.” 

Zakariya said: Mus‘ab said: I have forgotten the tenth, but it may have been rinsing the mouth with water.  

Imam an-Nawawi (may Allah have mercy on him) said: 

With regard to the fitrah, there was a difference of scholarly opinion as to what is meant by it in this case. Abu Sulaymaan al-Khattaabi said: Most of the scholars are of the view that it is the Sunnah (the way). This was also mentioned by a number of scholars other than al-Khattaabi, who said: What it means is that it is part of the ways of the Prophets (blessings and peace of Allah be upon them). And it was said that it is part of the religion.

End quote from Sharh Muslim by an-Nawawi, 3/147 

Al-Bukhaari (5892) and Muslim (259) narrated from Ibn ‘Umar (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: “Be different from the mushrikeen: let your beards grow and trim your moustaches.” 

“Let your beards grow” means: leave them alone. Fath al-Baari, 10/350. 

A matter like this is part of the religion, the Sunnah and the guidance of the Prophets. The individual should explain the ruling and present the evidence for it, even if the one who is asking the question is a kaafir. 

Muslim narrated in his Saheeh (386) that Salmaan al-Faarisi (may Allah be pleased with him) said: The mushrikoon said to us [according to Ahmad and others: One of the mushrikeen said to him when they were making fun of him]: I think that your companion is teaching you everything; he is even teaching you how to defecate!

Salmaan said: Indeed, he has told us not to clean ourselves using the right hand and not to face towards the qiblah (when defecating), and he has told us not to use dried dung or bones, and he said: “Do not clean yourselves using less than three stones.” 

See how this noble Sahaabi responded in this case having to do with the etiquette of relieving oneself and going to the toilet, without feeling any shame or embarrassment. 

Assuming that the one who asked him this question was mocking him, he turned the tables on him, as they say, and turned it into an opportunity to explain Islam to him. 

And Allah knows best.