Когда свет слишком яркий, я начинаю чаще моргать.

Here it means when in a general, repeated sense, so in natural English it is often close to whenever.

• Когда свет слишком яркий, я начинаю чаще моргать.
• = When/Whenever the light is too bright, I start blinking more often.

So this is not about one single moment only; it describes what usually happens in that situation.

Because Когда свет слишком яркий is a subordinate clause introduced by когда, and it comes before the main clause:

• Когда свет слишком яркий, = subordinate clause
• я начинаю чаще моргать. = main clause

Russian normally separates this kind of clause with a comma, just like English often does with When... , ....

In the present tense, Russian usually leaves out the verb to be.

So:

• свет слишком яркий literally looks like
• the light too bright

but it means

• the light is too bright

This is completely normal in Russian. You do not need есть here.

Because яркий is an adjective describing the noun свет.

• свет = light
• яркий = bright

They agree with each other:

• masculine
• singular
• nominative

So свет слишком яркий means the light is too bright.

By contrast, ярко is usually an adverb or a predicative word:

• Здесь слишком ярко. = It’s too bright here.

That version is also possible in Russian, but it is a different structure because there is no noun свет in it.

Because свет is the subject of the clause.

In свет слишком яркий:

• свет = the subject
• яркий = the predicate adjective agreeing with the subject

So nominative is exactly what you would expect here.

Слишком means too or excessively.

It modifies яркий:

• яркий = bright
• слишком яркий = too bright

So it works very much like English too in too bright.

Because this sentence describes a usual or repeated reaction, not one single future event.

• я начинаю is imperfective present
• it means I begin / I start
• in this context: I start doing this whenever that happens

If you said я начну, that would be perfective future:

• я начну = I will start

That would sound more like one specific future occasion, not a general habit.

Because after начинать / начать Russian normally uses an infinitive:

• начинать моргать = to start blinking
• начать читать = to start reading
• начать говорить = to start speaking

So:

• я начинаю моргать = I start blinking

This is a very common verb pattern in Russian.

Чаще is the comparative form of часто.

• часто = often
• чаще = more often

So:

• моргать часто = blink often
• моргать чаще = blink more often

In this sentence, чаще suggests an increase in blinking frequency because of the bright light.

Yes, and this is an important distinction.

• моргать = to blink (with your eyes)
• мигать = to flash, blink, flicker (like a light), and sometimes to wink

So for normal eye blinking, моргать is the right verb.

Examples:

• Я моргаю. = I am blinking.
• Лампа мигает. = The lamp is flashing/flickering.

Yes. Russian word order is flexible.

For example, you could also say:

• Я начинаю чаще моргать, когда свет слишком яркий.
• Я начинаю моргать чаще, когда свет слишком яркий.

These versions mean basically the same thing. The original sentence puts the when-clause first, which gives the condition or situation first:

• Когда свет слишком яркий, ...

That often sounds very natural when you want to set the scene before describing the result.

A helpful stress-marked version is:

• Когда́ свет сли́шком я́ркий, я начина́ю ча́ще морга́ть.

A few useful points:

• когда́ — stress on the last syllable
• сли́шком — stress on the first syllable
• я́ркий — stress on я
• начина́ю — stress on на
• ча́ще — stress on the first syllable
• морга́ть — stress on the last syllable

Pay special attention to чаще: the щ sound is softer and longer than English sh.