What they're suggesting is that your servo is trying to control the laser frequency but ends up getting confused by fluctuations in the laser output power.
For example, if your servo locks when the transmission photodiode reads 1V, but then your laser power drops to 80%, then your transmission photodiode will read 0.8V (so your error signal will be -0.2V), and the servo will compensate (even though the laser frequency had not changed).
What they suggest is to stop side-locking your laser and switch to a lock-in scheme. I drew a quick sketch below.
View attachment 317038
The key here is that your error signal lockpoint will become independent of laser power. If you want a mathematical argument for why this is, I would recommend reading about the math behind PDH locking. The main difference between the above scheme and PDH locking is that this scheme uses the transmission photodiode (PDH typically uses reflection photodiode, for higher bandwidth) and the strength of the modulating signal.
As far as jargon, "dither" usually means a small amount of frequency modulation used to generate the error signal, and "lock-in detection" likely means using the demodulated photodiode signal instead of the DC photodiode signal as your error signal. Were there other terms that were confusing?
P.S. It only just occurred to me that because you've been locking with the transmission photodiode, your servo never had a chance of dealing with the high frequency noise on your error signal. The transmitted power out of the cavity is band-limited by the cavity linewidth, so if your linewidth is 30kHz, you will never be able to stabilize noise faster than 30kHz while locking to the transmission photodiode.
If you want to go faster than that, you need to use the reflection signal (which is not limited by the cavity linewidth). Sorry it took me so long to put that together!
Edit: If you're looking for a good place to learn about PDH locking, I suggest Holger Mueller's "Electronics for Pros" lecture #18,
see here.
Edit edit: Actually, I'm not sure if the reflection signal on a bowtie cavity is bandlimited by the linewidth or not (it definitely is not limited for Fabry-Perot cavities, but I don't know about bowtie cavities). I added strikethrough to my comments where I made this claim. Sorry!