The athar attributed to Umar was documented in Sharh Zād al-Mustaqni' by Mohammad al-Mukhtār ash-Sahanqīti. It is about when 'Umar was judging a case in court. When he told one of the witnesses that he did not know him, a man among the attendees rose and said that he knew the witness, so 'Umar questioned him:
فجاءه برجل فقال عمر للرجل: أتعرفه؟ قال: أعرفه بالعدالة والأمانة. قال: أهو جارك الذي تعرف مدخله ومخرجه؟ قال: لا. قال: أعاملته بالدينار والدرهم الذي يعرف به صدق الرجل من كذبه؟ قال: لا. قال: أسافرت معه؟ قال: لا. قال: لا تعرفه. ثم قال للرجل: اذهب وائتني بمن يعرفك.
— NOTE: My own translation, so treat with care.
A man came forward, so 'Umar asked him [about the witness]: "Do you known him [the witness]?" The man answered: "I know his trustworthiness and honesty."
'Umar asked him: "Is he your neighbor that you observe when he goes in and out?" He answered: "No."
'Umar asked the man: "Did you deal with him in dinārs and dirhams which discerns a man's honesty from dishonesty?" He answered: "No."
'Umar asked the man: "Did you travel with him?" He answered: "No."
'Umar told the man: "You do not know him." Then 'Umar told the other man [the witness]: "Go get someone who knows you."
— Sharh Zād al-Mustaqni', Vol. 65, pp. 3 (Arabic only)
The story is in numerous other sources either with similar wording (e.g., in Ihyā' 'Ulūm ad-Dīn 3/160 by Abu Hāmid al-Ghazali), or with slightly different wording (e.g., in As-Samt 274 by Ibn Abi ad-Dunya, which was quoted by Al-Hindi in Kanz al-'Ummāl 9/173 and by As-Suyūti in Jāmi' al-Ahādīth 28/331, and in Al-Balāgha al-'Umariyyah 191 by Mohammad Sālem al-Khadr). It is also in other books with some other additional questions (e.g., in Al-Mashyakha al-Baghdādiyyah 2/54 by Abu Tāhir as-Salafi).
This athar has three different narration chains; this is why you may find that it was graded authentic by some scholars (e.g., Abu 'Ali Sa'īd ibn 'Uthmān ibn as-Sakan), leaning more toward authentic (e.g., Mohammad Nasiruddīn Al-Albāni), or weak by others (e.g., Abu Ja'far al-'Aqīli). There are a few lessons learned from this athar:
Neighbors — at the time, at least — knew each other quite well, and they observed the rights of the neighborhood (see Sahih al-Bukhari 90/28, Sahih Muslim 45/184, Sunan Abi Dawud 24/103, etc.). Their interactions showed if they observed Islamic teachings on the matter (see Sahih al-Bukhari 78/48, Sahih Muslim 1/82, Sahih al-Bukhari 46/24, etc.). In general, the proximity demonstrated their behavioral pattern over a period of time, especially when the neighbors are physically close (denoted by being able to see him going in and out).
Financial dealings typically demonstrate the honesty (or dishonesty) of a person (see Qur'an 3:75 about being entrusted with small or large sums of money, and Sahih al-Bukhari 92/37 where the Prophet talks about people carrying out a trade without trustworthiness). There is no need to elaborate on this one as we — as humans — know exactly what the significance of money is to the majority of the human race.
Isma'īl al-'Ajlūni said in Kasfh al-Khafā' (1/518) that travel reveals men's manners (Arabic: السفر يسفر عن أخلاق الرجال). The word safar (Arabic: سفر) means, in addition to travel, to expose or to reveal or to make clear. You will find many Islamic pieces of literature dedicating chapters on how travel shows the real manners of people when they are away from their own society with no peer pressure or social pressure, and where adaptability to foreign conditions and resilience are required and are demonstrated through interactions with their hosts (see Sharh Zād al-Mustaqni' 3/65, Arabic only).
The trustworthiness of witnesses is a requirement, and such trustworthiness has to be an obvious and known attribute of the person providing a testimony (see Qur'an 65:2). If this trustworthiness is not known or obvious to the judge, the judge should not accept an unknown person and must get someone who knows the witness to vouch for the witness's trustworthiness (which is what 'Umar was doing by questioning the man who brought forth the witness). Note that this is not the general rule; rather, the general rule is that a Muslim is trustworthy for the purpose of testimony unless proven otherwise as 'Umar himself said (see Sharh Zād al-Mustaqni' 7/340). The validation of the trustworthiness of the witness in the case of this athar was for a doubt that 'Umar might have had, so he was doing the mandatory investigation (see Qur'an 49:6).
The applicability of this athar is still valid until today, although its application may require adaptation to our current societies.