how is the rsa $publickey(n ,e)$ transformed into a textfile allowing its public distribtution? wikipedia says about the keys itself:


  The public key consists of the modulus n and the public (or
  encryption) exponent e. The private key consists of the modulus n and
  the private (or decryption) exponent d, which must be kept secret.


I know, that public keys look like that in the wild:


  -----BEGIN RSA PUBLIC KEY----- 
  MIIBCgKCAQEA+xGZ/wcz9ugFpP07Nspo6U17l0YhFiFpxxU4pTk3Lifz9R3zsIsu
  ERwta7+fWIfxOo208ett/jhskiVodSEt3QBGh4XBipyWopKwZ93HHaDVZAALi/2A
  +xTBtWdEo7XGUujKDvC2/aZKukfjpOiUI8AhLAfjmlcD/UZ1QPh0mHsglRNCmpCw mwSXA9VNmhz+PiB+Dml4WWnKW/VHo2ujTXxq7+efMU4H2fny3Se3KYOsFPFGZ1TN
  QSYlFuShWrHPtiLmUdPoP6CV2mML1tk+l7DIIqXrQhLUKDACeM5roMx0kLhUWB8P
  +0uj1CNlNN4JRZlC7xFfqiMbFRU9Z4N6YwIDAQAB
  
  -----END RSA PUBLIC KEY-----


or sth like:


   Subject Public Key Info:
       Public Key Algorithm: rsaEncryption
       RSA Public Key: (1024 bit)
           Modulus (1024 bit):
               00:b4:31:98:0a:c4:bc:62:c1:88:aa:dc:b0:c8:bb:
               33:35:19:d5:0c:64:b9:3d:41:b2:96:fc:f3:31:e1:
               66:36:d0:8e:56:12:44:ba:75:eb:e8:1c:9c:5b:66:
               70:33:52:14:c9:ec:4f:91:51:70:39:de:53:85:17:
               16:94:6e:ee:f4:d5:6f:d5:ca:b3:47:5e:1b:0c:7b:
               c5:cc:2b:6b:c1:90:c3:16:31:0d:bf:7a:c7:47:77:
               8f:a0:21:c7:4c:d0:16:65:00:c1:0f:d7:b8:80:e3:
               d2:75:6b:c1:ea:9e:5c:5c:ea:7d:c1:a1:10:bc:b8:
               e8:35:1c:9e:27:52:7e:41:8f



=> My question is, how are those numbers $n$ and $e$ transformed to a keyfile like above?



EDIT: Additionally, i was specifically wondering how the numbers are calculated into a keyfile. I mean, if I had a single really big number, i can imagine the number is easily converted into one of your encoding. But the public key is a combination of $n$ and $e$, and i don't know which combination.
There are two concepts that are relevant for your question: Base64 encoding and ASN.1 DER encoding. 

So, to deal with the most immediate concern, take n and e as the representation of the RSA values stored in a byte array. 

Base 64

Given that cryptographic algorithms usually deal with purely binary data (keys, ciphertexts, random values), their representation can lead to communication issues. The most evident: you can't send the key as a string (e.g. via email) because it can be misinterpreted and even reformatted. What is just a random byte of a key could be an EOL character, and parsing can be jeopardized.

Base64 aims to solve all that family of issues by encoding the data using only 64 printable characters. I will not detail here the encoding rules, but, at a cost of roughly 30% of communication overhead, you get rid of encoding issues dependent on the platform. A base 64 string will have only letters, numbers and the symbols "+", "/" and "=". 

ASN.1 DER

The Distinguished Encoded Rules define the encoding of certain data types with the format TLV: Type Length Value. Some of the types can be classical (BOOLEAN, INTEGER), some can be understandable (BIT STRING, OCTET STRING, UTF8String, UTCTIme), some are rather specific (OBJECT IDENTIFIER, SEQUENCE, SET) and some are constructions that build over existing data types to build upon. 

Unless you're very interested in the details of how it works, I would suggest to avoid the details about the encoding of each data type and use a parser instead. The encoding rules are a bit dark and contribute little to nothing to the understanding of what's built on top. You can, however, Base64-decode the public key you posted and pass the result to a tool like dumpasn1 to see how nested the structure can get.

What you're seeing

Given all the background, I can finally reply to your question. The first key you got is a PEM encoding: a ASN.1 DER structure subsequently Base-64. I have seen the structure below as an equivalent and functional text representation of the same data, but I can't seem to find the right name for that encoding.



UPDATE: It seems I published an incomplete version of my answer. Particularly, I did not include the link to the RFC 3280, which contains the definition of the DER encoding of a RSA public key (page 15). Thanks to @dave_thompson_085 for bringing it up and the useful additional links:


RFC 3280 (page 15)
PEM
RFC 3447 (Appendix A)
Wikipedia ASN.1 entry
Wikipedia BER encoding entry