The Prophet, sallallaahu alayhi wa sallam, said: He is not of us who is not kind to our youngesters, and shows no respect to our elders....
Please explain this Hadith.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

The narration which you mentioned in the question is reported by At-Tirmithi, Abu Daawood, Ahmad and others and it is an authentic narration. The wording of At-Tirmithi is as follows: "He is not one of us who is not merciful to our youngsters and does not respect and honour our elders."  The wording of Abu Daawood is as follows: "He, who is not merciful to our youngsters and does not fulfil the rights of our elderly, is not one of us."

The meaning of the above narration is that he who is not merciful to the young people and does not respect and honour the elders and fulfil their rights, is not following the Prophet's methodology and is not acting according to his guidance. The term "elders" include the young men and older people, so whoever is not merciful to the young by being affectionate and kind to them, and does not fulfil the right of the elders, and does not honour and respect them, then he is not upon the guidance, the way and the Sunnah of the Prophet .  

Allaah Knows best.