Where does "Whatcha" & "Didja" come from?

These words are the transcriptions of a particular pronunciation of two words and they represent really two words but somehow one came to represent two and three words (whatcha). The origin of this pronunciation is explained mostly phonetically; it is a phenomenon better understood through an identification of the phonetic processes involved.

Longman Pronunciation Dictionary (pp. 49,50):

Assimilation is a kind of COARTICULATION. It is the alteration of a speech sound to make it sound more similar to its neighbours. In English it mainly affects PLACE OF ARTICULATION.

Yod coalescence (or 'coalescent' assimilation) is the process which changes /t/ or /d/ plus /j/ into /tʃ/ or /dʒ/ respectively. Across word boundaries it mainly affects phrases involving "you" or "your".

Longman Pronunciation Dictionary (p.151) Coarticulation:

Speech sounds tend to be influenced by the speech sounds that surround them. Coarticulation is the retention of a phonetic feature that was present in a preceding sound, or the anticipation of a feature that will be needed for a following sound. MOST ALLOPHONIC variation — though not all — is coarticulatory.

It becomes clearer now why the word "you" is found in both combinations; however there is a gap to fill in explaining how the forms in which a separating word occurs, also came to have the same pronunciation.

From Wiktionary:

whatcha ( watcha, wotcha, wotcher, whatchya)
(colloquial) Contraction of "what are you".

• Whatcha doin'?
• Whatcha talking 'bout?
• Whatcha sayin'?

(colloquial) Contraction of "what have you".

• Whatcha been up to?

(colloquial) Contraction of "what do you".

• Whatcha think?
• Whatcha got there?
• Whatcha wanna do now?

(colloquial) Contraction of "what you".

• Do whatcha gotta do!

didja (informal) Pronunciation spelling of "did you". [Wiktionary]

Whatcha and didja are the 'informal' spellings of what you and did you respectively. People often spell them the way they pronounce these words.
'Palatalisation' is what caused those pronunciations.

Palatalisation is a type of secondary articulation whereby a sound is produced with the front part of the tongue coming near the hard palate.
It's often regressive and is influenced by the following /j/.
In consonants, it mostly affects the alveolar obstruents (/t d s z/) when they're immediately followed by /j/, resulting in palatalised sounds. In IPA, the palatalised sounds are usually transcribed by adding a superscript [^j] after the primary symbol.

Most secondary articulations are allophonic in English i.e. they do not change the meaning of the word. For example, 'tune' is usually pronounced [t͡ʃuːn] in most varieties of British English while most Americans pronounce it [tuːn], both of them mean the same thing.
In some languages, such as Russian, secondary articulations change the meaning of the word, thereby making phonemic contrasts. For example, [s^jok] means 'he lashed' while [sok] means 'juice'. _{[Example from Understanding Phonetics by Patricia Ashby]}

In English,

• /t/ and /j/ coalesce into /t͡ʃ/
• /d/ and /j/ coalesce into /d͡ʒ/
• /s/ and /j/ coalesce into /ʃ/
• /z/ and /j/ coalesce into /ʒ/

Most of time, the clusters /tr/ and /dr/ also undergo palatalisation and become /t͡ʃr/ and /d͡ʒr/, respectively.
So 'true' and 'drive' might sound as if they were 'chru' and 'jrive'.

Examples:

• What you
• Did you
• Tune (in BrE)
• Gradual
• Nature etc.
• Fusion

Words like situation, gradual, perpetual, Tuesday, education etc., have also undergone palatalisation.

Geoff Lindsey in his book English after RP explains that this coalescence/palatalisation had already happened to some words by the time of RP, e.g. culture, future, picture, and soldier, so this is not a random idiosyncrasy.

_{/j/ represents the 'y' sound as in yes.
/ʃ/ is the 'sh' as in ship.
/ʒ/ as in genre.
/t͡ʃ/ as in church.
/d͡ʒ/ as in judge.}

This feature of connected speech is often called 'yod coalescence'.

Whatcha (or wotcha) is a contraction of a variety of expressions: what are/do/did/have you - see the Wiktionary entry.

Similarly didja.

They arose because that's how people speak.