The "I am" (in the English translation) is implied by the conjugation of the verb (as said in ruben2020's answer).
In Pali, verbs are "conjugated" -- i.e. their endings change to denote "which person?" and "how many?" (as well as, "which tense?").
That's true of other languages (e.g. Latin):
- amo ("I love")
- amas ("you love")
- amat ("he/she loves")
In English verbs don't conjugate, instead they're nearly invariant and instead there's a separate personal pronoun, e.g. ...
- I run
- you run
- he run[s]
- we run
- you run
- they run
... i.e. you get the person and the number from the added pronoun, not from conjugating the verb-ending. Alternatively ...
- I am running
- you are running
In languages which conjugate it's ungrammatical not to (conjugation is required) -- i.e. I expect that you can't (with good grammar) use a verb in a Pali sentence without conjugating it.
The conjugation is just "conventional" though -- maybe you shouldn't read too much into it -- and it is NOT meant to mean ...
I am thinking. Oh my God: 'I am' thinking. So, "I am"! ... "I AM!"
As an aside, when Ven. Yuttadhammo for example describes "noting" something in meditation, he uses the present participle form -- e.g. "Breathing ... breathing", rather than, "I am breathing, I am breathing". This (i.e. without the personal pronoun, without the whole "I am" part of the so-called present continuous phrase) isn't ordinarily/conventionally grammatical (to be grammatical, a sentence usually requires a subject as well as a verb), but is maybe better suited to the occasion and purpose (i.e. of noting but without wanting to reinforce a self-view).
If I look at the French translation of the sutta, I see "Je marche", which translates to English as either "I walk" or "I am walking".
I don't know the Sinhalese language but I think Wikipedia says it doesn't conjugate:
Sinhalese distinguishes three conjugation classes. Spoken Sinhalese does not mark person, number or gender on the verb (literary Sinhalese does). In other words, there is no subject–verb agreement.
Maybe that's why there is no equivalent in the Sinhalese translation -- i.e. the grammar doesn't require it.