This whole mantra forms the first passage of Taittariya Aranyaka of YajurVeda. The first verse "Om Bhadram Karnebhi Srinuyaam Devaha" also exists in RigVeda 1.89.8
Here is the meaning of the above mantra:
ॐ भद्रं कर्णेभिः शृणुयाम देवाः ।
Om Bhadram Karnebhiḥ Srinuyaam Devah
Om, May we hear what is Auspicious with our ears,Oh Gods
भद्रं पश्येमाक्षभिर्यजत्राः
Bhadram Pashyem Akshabhir Yajatrah
May we see what is Auspicious With our eyes, Oh holy ones worthy of Worship
स्थिरैरङ्गैस्तुष्टुवागँसस्तनूभिः
Sthirair Angais Tushtuvagamsas Tanoobhih
May we live a life of satisfaction with strong organs and Healthy body
व्यशेम देवहितं यदायूः
Vyashema Devahitam Yadaayooh
May we Praise the Lord during the life span given to us by the Gods
स्वस्ति न इन्द्रो वृद्धश्रवाः
Swasti Na Indro Vriddhashravah
May God Indra of Great fame bless us
स्वस्ति नः पूषा विश्ववेदाः
Swasti Na Poosha Vishwa-Vedaah
May the Omniscient Poosha bless us
स्वस्ति नस्ताक्षर्यो अरिष्टनेमिः
Swasti Nastaaksharyo Arishtanemih
May the Protector Garuda bless us
स्वस्ति नो ब्रुहस्पतिर्दधातु
Swasti No Brihaspatir-dadhaatu
May Lord Brihaspati protect us
ॐ शान्तिः शान्तिः शान्तिः
Om, Shaanti Shaanti Shaanti
Om, Peace, Peace, Peace
Hence in a single paragraph the mantra means:
Om, May we hear what is Auspicious with our ears,Oh Gods. May we see what is Auspicious With our eyes, Oh holy ones worthy of Worship. May we live a life of satisfaction with strong organs and Healthy body. May we Praise the Lord during the life span given to us by the Gods. May God Indra of Great fame bless us. May the Omniscient Poosha bless us. May the Protector Garuda bless us. May Lord Brihaspati protect us. Om, Peace, Peace, Peace
Refrences used from here.