Think of the 19th century like a cave in a mountain. The mouth of the cave is like the year 1800. The deepest part of the cave is like the year 1899. If you go "into" the cave, it means you have walked through the mouth, but it is not clear how far you went. If you go "well into" the cave, it means you have traveled a significant distance inside. This is the sense or feeling of the phrase "well into". So, "well into the 19th century" means that starting from the year 1800 and for a significant period of time afterwards (say, 30 or 50 years), penalties consisted primarily of public humiliation. Please note that this sentence also implies that "penalties consisted primarily of public humiliation" before the year 1800 also. Finally, it is technically incorrect to say that the 19th century began in the year 1800--it actually began in the year 1801. However, many English speakers do not know this, so you may sound funny if you insist on using the correct year.

until the 19th century = until the year 1800 until well into the 19th century = until around 1830-1850 or so